{"id":3018,"date":"2026-01-21T12:13:59","date_gmt":"2026-01-21T06:43:59","guid":{"rendered":"https:\/\/www.manabadi.co.in\/boards\/?p=3018"},"modified":"2026-01-21T16:50:59","modified_gmt":"2026-01-21T11:20:59","slug":"ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5c","status":"publish","type":"post","link":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5c\/","title":{"rendered":"TS Inter 1st Year Maths 1A Products of Vectors Solutions Exercise 5(C)"},"content":{"rendered":"\n<h3>I.<br> Question 1.<br> Compute [i\u0305 \u2013 j\u0305 j\u0305 \u2013 k\u0305 k\u0305 \u2013 i\u0305]<\/h3>\n\n<p>Answer:<br> [i\u0305 \u2013 j\u0305 j\u0305 \u2013 k\u0305 k\u0305 \u2013 i\u0305] = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-1-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1\" style=\"width: 7.831em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.759em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1006.62em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2\"><span class=\"mrow\" id=\"MathJax-Span-3\"><span class=\"mo\" id=\"MathJax-Span-4\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-5\"><span style=\"display: inline-block; position: relative; width: 5.827em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-6\"><span class=\"mrow\" id=\"MathJax-Span-7\"><span class=\"mn\" id=\"MathJax-Span-8\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-16\"><span class=\"mrow\" id=\"MathJax-Span-17\"><span class=\"mn\" id=\"MathJax-Span-18\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-26\"><span class=\"mrow\" id=\"MathJax-Span-27\"><span class=\"mo\" id=\"MathJax-Span-28\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-29\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.107em, -999.998em); top: -4.471em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-9\"><span class=\"mrow\" id=\"MathJax-Span-10\"><span class=\"mo\" id=\"MathJax-Span-11\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-12\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-19\"><span class=\"mrow\" id=\"MathJax-Span-20\"><span class=\"mn\" id=\"MathJax-Span-21\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-30\"><span class=\"mrow\" id=\"MathJax-Span-31\"><span class=\"mn\" id=\"MathJax-Span-32\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.06em, -999.998em); top: -4.471em; left: 4.569em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-13\"><span class=\"mrow\" id=\"MathJax-Span-14\"><span class=\"mn\" id=\"MathJax-Span-15\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-22\"><span class=\"mrow\" id=\"MathJax-Span-23\"><span class=\"mo\" id=\"MathJax-Span-24\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-25\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-33\"><span class=\"mrow\" id=\"MathJax-Span-34\"><span class=\"mn\" id=\"MathJax-Span-35\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-36\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-1\">\\left|\\begin{array}{rrr}  1 & -1 & 0 \\\\  0 & 1 & -1 \\\\  -1 & 0 & 1  \\end{array}\\right|<\/script><br> = 1 (1) + 1 (- 1) = 1 \u2013 1 = 0<\/p>\n\n<h3>Question 2.<br> If a\u0305 = i\u0305 \u2013 2j\u0305 \u2013 3k\u0305, b\u0305 = 2i\u0305 + j\u0305 \u2013 k\u0305, c\u0305 = i\u0305 + 3j\u0305 \u2013 2k\u0305 then compute a\u0305 . (b\u0305 \u00d7 c\u0305)<\/h3>\n\n<p>Answer:<br> Given a\u0305 = i\u0305 \u2013 2j\u0305 \u2013 3k\u0305, b\u0305 = 2i\u0305 + j\u0305 \u2013 k\u0305, c\u0305 = i\u0305 + 3j\u0305 \u2013 2k\u0305 then<br> a\u0305.(b\u0305 \u00d7 c\u0305) = (a\u0305 b\u0305 c\u0305) = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-2-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-37\" style=\"width: 6.992em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.014em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.87em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-38\"><span class=\"mrow\" id=\"MathJax-Span-39\"><span class=\"mo\" id=\"MathJax-Span-40\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-41\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-42\"><span class=\"mrow\" id=\"MathJax-Span-43\"><span class=\"mn\" id=\"MathJax-Span-44\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-53\"><span class=\"mrow\" id=\"MathJax-Span-54\"><span class=\"mn\" id=\"MathJax-Span-55\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-63\"><span class=\"mrow\" id=\"MathJax-Span-64\"><span class=\"mn\" id=\"MathJax-Span-65\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.107em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-45\"><span class=\"mrow\" id=\"MathJax-Span-46\"><span class=\"mo\" id=\"MathJax-Span-47\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-48\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-56\"><span class=\"mrow\" id=\"MathJax-Span-57\"><span class=\"mn\" id=\"MathJax-Span-58\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-66\"><span class=\"mrow\" id=\"MathJax-Span-67\"><span class=\"mn\" id=\"MathJax-Span-68\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.26em, 6.247em, -999.998em); top: -4.564em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-49\"><span class=\"mrow\" id=\"MathJax-Span-50\"><span class=\"mo\" id=\"MathJax-Span-51\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-52\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-59\"><span class=\"mrow\" id=\"MathJax-Span-60\"><span class=\"mo\" id=\"MathJax-Span-61\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-62\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-69\"><span class=\"mrow\" id=\"MathJax-Span-70\"><span class=\"mo\" id=\"MathJax-Span-71\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-72\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-73\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-2\">\\left|\\begin{array}{ccc}  1 & -2 & -3 \\\\  2 & 1 & -1 \\\\  1 & 3 & -2  \\end{array}\\right|<\/script><br> = 1 (- 2 + 3) + 2 (- 4 + 1) \u2013 3 (6 \u2013 1)<br> = 1 \u2013 6 \u2013 15 = \u2013 20<\/p>\n\n<h3>Question 3.<br> If a\u0305 = (1, -1, -6), b\u0305 = (1, -3, 4) and c\u0305 = (2, -5, 3), then compute the following.<br> (i) a\u0305 . (b\u0305 \u00d7 c\u0305)<\/h3>\n\n<p>Answer:<br> a\u0305 \u00d7 (b\u0305 \u00d7 c\u0305) = (a\u0305 . c\u0305) b\u0305 \u2013 (a\u0305 . b\u0305) c\u0305<br> = (2 + 5 \u2013 18) b\u0305 \u2013 (1 + 3 \u2013 24) c\u0305<br> = -11b\u0305 + 20c\u0305<br> = \u2014 11 (i\u0305 \u2013 3j\u0305 + 4k\u0305) + 20(2i\u0305 \u2013 5j\u0305 + 3k\u0305)<br> = 29i\u0305 \u2013 67j\u0305 + 16k\u0305<\/p>\n\n<p>(ii) a\u0305 \u00d7 (b\u0305 \u00d7 c\u0305)<br> Answer:<br> a\u0305 \u00d7 (b\u0305 \u00d7 c\u0305) = (a\u0305.c\u0305)b\u0305 \u2013 (a\u0305.b\u0305)c\u0305<br> = (2 + 5 \u2013 18)b\u0305 \u2013 (1 + 3 \u2013 24)c\u0305<br> = -11b\u0305 + 20c\u0305<br> = -11(i\u0305 \u2013 3j\u0305 + 4k\u0305) + 20(2i\u0305 \u2013 5j\u0305 + 3k\u0305)<br> = 29i\u0305 \u2013 67j\u0305 + 16k\u0305<\/p>\n\n<p>iii) (a\u0305 \u00d7 b\u0305) \u00d7 c\u0305<br> Answer:<br> (a\u0305 \u00d7 b\u0305) \u00d7 c\u0305<br> = (a\u0305 . c\u0305)b\u0305 \u2013 (b\u0305 . c\u0305)a\u0305<br> = (2 + 5 \u2013 18) b\u0305 \u2013 (2 + 15 + 12) a\u0305<br> = -11 (i\u0305 \u2013 3j\u0305 + 4k\u0305) \u2013 29 (i\u0305 \u2013 j\u0305 \u2013 6k\u0305)<br> = -40i\u0305 + 62j\u0305 + 130k\u0305<\/p>\n\n<h3>Question 4.<br> Simplify the following :<br> i) (i\u0305 \u2013 2j\u0305 + 3k\u0305) \u00d7 (2i\u0305 + j\u0305 \u2013 k\u0305) \u2013 (j\u0305 + k\u0305)<\/h3>\n\n\n\n<p>Answer:<br> (i\u0305 \u2013 2j\u0305 + 3k\u0305) \u00d7 (2i\u0305 + j\u0305 \u2013 k\u0305) \u2013 (j\u0305 + k\u0305)<br> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-3-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-74\" style=\"width: 6.992em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.014em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.87em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-75\"><span class=\"mrow\" id=\"MathJax-Span-76\"><span class=\"mo\" id=\"MathJax-Span-77\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-78\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-79\"><span class=\"mrow\" id=\"MathJax-Span-80\"><span class=\"mn\" id=\"MathJax-Span-81\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-89\"><span class=\"mrow\" id=\"MathJax-Span-90\"><span class=\"mn\" id=\"MathJax-Span-91\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-99\"><span class=\"mrow\" id=\"MathJax-Span-100\"><span class=\"mn\" id=\"MathJax-Span-101\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.06em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-82\"><span class=\"mrow\" id=\"MathJax-Span-83\"><span class=\"mo\" id=\"MathJax-Span-84\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-85\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-92\"><span class=\"mrow\" id=\"MathJax-Span-93\"><span class=\"mn\" id=\"MathJax-Span-94\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-102\"><span class=\"mrow\" id=\"MathJax-Span-103\"><span class=\"mn\" id=\"MathJax-Span-104\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.06em, -999.998em); top: -4.471em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-86\"><span class=\"mrow\" id=\"MathJax-Span-87\"><span class=\"mn\" id=\"MathJax-Span-88\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-95\"><span class=\"mrow\" id=\"MathJax-Span-96\"><span class=\"mo\" id=\"MathJax-Span-97\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-98\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-105\"><span class=\"mrow\" id=\"MathJax-Span-106\"><span class=\"mn\" id=\"MathJax-Span-107\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-108\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-3\">\\left|\\begin{array}{rrr}  1 & -2 & 3 \\\\  2 & 1 & -1 \\\\  0 & 1 & 1  \\end{array}\\right|<\/script><br> = 1 (2) + 2(2) + 3(2)<br> = 2 + 4 + 6<br> = 12<\/p>\n\n<p>ii) (2i\u0305 \u2013 3j\u0305 + k\u0305) \u2013 (i\u0305 \u2013 j\u0305 + 2k\u0305) \u00d7 (2i\u0305 + j\u0305 + k\u0305)<br> Answer:<br> (2 i\u0305 \u2013 3j\u0305 + k\u0305) . (i\u0305 \u2013 j\u0305 + 2k\u0305) \u00d7 (2i\u0305 + j\u0305 + k\u0305)<br> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-4-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-109\" style=\"width: 6.06em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.221em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.08em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-110\"><span class=\"mrow\" id=\"MathJax-Span-111\"><span class=\"mo\" id=\"MathJax-Span-112\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-113\"><span style=\"display: inline-block; position: relative; width: 4.289em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-114\"><span class=\"mrow\" id=\"MathJax-Span-115\"><span class=\"mn\" id=\"MathJax-Span-116\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-124\"><span class=\"mrow\" id=\"MathJax-Span-125\"><span class=\"mn\" id=\"MathJax-Span-126\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-134\"><span class=\"mrow\" id=\"MathJax-Span-135\"><span class=\"mn\" id=\"MathJax-Span-136\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.26em, 6.06em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-117\"><span class=\"mrow\" id=\"MathJax-Span-118\"><span class=\"mo\" id=\"MathJax-Span-119\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-120\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-127\"><span class=\"mrow\" id=\"MathJax-Span-128\"><span class=\"mo\" id=\"MathJax-Span-129\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-130\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-137\"><span class=\"mrow\" id=\"MathJax-Span-138\"><span class=\"mn\" id=\"MathJax-Span-139\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-121\"><span class=\"mrow\" id=\"MathJax-Span-122\"><span class=\"mn\" id=\"MathJax-Span-123\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-131\"><span class=\"mrow\" id=\"MathJax-Span-132\"><span class=\"mn\" id=\"MathJax-Span-133\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-140\"><span class=\"mrow\" id=\"MathJax-Span-141\"><span class=\"mn\" id=\"MathJax-Span-142\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-143\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-4\">\\left|\\begin{array}{rrr}  2 & -3 & 1 \\\\  1 & -1 & 2 \\\\  2 & 1 & 1  \\end{array}\\right|<\/script><br> = 2 (- 1 \u2013 2) + 3 (1 \u2013 4) + 1 (1 + 2)<br> = \u2013 6 \u2013 9 + 3<br> = -12<\/p>\n\n<h3>Question 5.<br> Find the volume of the parallelopiped having coterminus edges i\u0305 + j\u0305 + k\u0305, i\u0305 \u2013 j\u0305 and i\u0305 + 2j\u0305 \u2013 k\u0305<\/h3>\n\n<p>Answer:<br> Let a\u0305 = i\u0305 + j\u0305 + k\u0305, b\u0305 = i\u0305 \u2013 j\u0305 and c\u0305 = i\u0305 + 2j\u0305 \u2013 k\u0305 then the volume of parallelopiped =<br> | (a\u0305 b\u0305 c\u0305)|<br> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-5-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-144\" style=\"width: 6.992em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.014em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.87em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-145\"><span class=\"mrow\" id=\"MathJax-Span-146\"><span class=\"mo\" id=\"MathJax-Span-147\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-148\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.42em, 6.06em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-149\"><span class=\"mrow\" id=\"MathJax-Span-150\"><span class=\"mn\" id=\"MathJax-Span-151\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-158\"><span class=\"mrow\" id=\"MathJax-Span-159\"><span class=\"mn\" id=\"MathJax-Span-160\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-168\"><span class=\"mrow\" id=\"MathJax-Span-169\"><span class=\"mn\" id=\"MathJax-Span-170\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.06em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-152\"><span class=\"mrow\" id=\"MathJax-Span-153\"><span class=\"mn\" id=\"MathJax-Span-154\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-161\"><span class=\"mrow\" id=\"MathJax-Span-162\"><span class=\"mo\" id=\"MathJax-Span-163\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-164\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-171\"><span class=\"mrow\" id=\"MathJax-Span-172\"><span class=\"mn\" id=\"MathJax-Span-173\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-155\"><span class=\"mrow\" id=\"MathJax-Span-156\"><span class=\"mn\" id=\"MathJax-Span-157\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-165\"><span class=\"mrow\" id=\"MathJax-Span-166\"><span class=\"mn\" id=\"MathJax-Span-167\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-174\"><span class=\"mrow\" id=\"MathJax-Span-175\"><span class=\"mo\" id=\"MathJax-Span-176\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-177\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-178\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-5\">\\left|\\begin{array}{rrr}  1 & 1 & 1 \\\\  1 & -1 & 0 \\\\  1 & 2 & -1  \\end{array}\\right|<\/script><br> = 1 (1) \u2013 1 (- 1) + 1 (2 + 1)<br> = 1 + 1 + 3 = 5 cubic units.<\/p>\n\n\n\n<h3>Question 6.<br> Find \u2018t\u2019 for which the vectors 2i\u0305 \u2013 3j\u0305 + k\u0305, i\u0305 + 2j\u0305 \u2013 3k\u0305 and j\u0305 \u2013 tk\u0305 are coplanar.<\/h3>\n\n<p>Answer:<br> Denote the given vectors by a, b, c .and if the vectors are coplanar then [a\u0305 b\u0305 c\u0305] = 0<br> \u21d2 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-6-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-179\" style=\"width: 6.992em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.014em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.87em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-180\"><span class=\"mrow\" id=\"MathJax-Span-181\"><span class=\"mo\" id=\"MathJax-Span-182\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-183\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-184\"><span class=\"mrow\" id=\"MathJax-Span-185\"><span class=\"mn\" id=\"MathJax-Span-186\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-194\"><span class=\"mrow\" id=\"MathJax-Span-195\"><span class=\"mn\" id=\"MathJax-Span-196\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-204\"><span class=\"mrow\" id=\"MathJax-Span-205\"><span class=\"mn\" id=\"MathJax-Span-206\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.26em, 6.06em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-187\"><span class=\"mrow\" id=\"MathJax-Span-188\"><span class=\"mo\" id=\"MathJax-Span-189\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-190\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-197\"><span class=\"mrow\" id=\"MathJax-Span-198\"><span class=\"mn\" id=\"MathJax-Span-199\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-207\"><span class=\"mrow\" id=\"MathJax-Span-208\"><span class=\"mn\" id=\"MathJax-Span-209\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.26em, 6.247em, -999.998em); top: -4.564em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-191\"><span class=\"mrow\" id=\"MathJax-Span-192\"><span class=\"mn\" id=\"MathJax-Span-193\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-200\"><span class=\"mrow\" id=\"MathJax-Span-201\"><span class=\"mo\" id=\"MathJax-Span-202\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-203\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1001.12em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-210\"><span class=\"mrow\" id=\"MathJax-Span-211\"><span class=\"mo\" id=\"MathJax-Span-212\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-213\" style=\"font-family: MathJax_Math-italic;\">t<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-214\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-6\">\\left|\\begin{array}{rrr}  2 & -3 & 1 \\\\  1 & 2 & -3 \\\\  0 & 1 & -t  \\end{array}\\right|<\/script> = 0<br> \u21d2 2 (- 2t + 3) + 3 (- t) + 1 (1) = 0<br> \u21d2 \u2013 7t + 7 = 0<br> \u21d2 t = 17.<\/p>\n\n<h3>Question 7.<br> For non coplanar vectors a\u0305,b\u0305 and c\u0305, determine p for which the vectors a\u0305 + b\u0305 + c\u0305, a\u0305 + pb\u0305 + 2c\u0305 and -a\u0305 + b\u0305 + c\u0305 are coplanar.<\/h3>\n\n<p>Answer:<br> Given a,b,c are non coplanar vectors We have [a\u0305 b\u0305 c\u0305] = 0 If the vectors a\u0305 + b\u0305 + c\u0305, a\u0305 + pb\u0305 + 2c\u0305 and -a\u0305 + b\u0305 + c\u0305 are coplanar.<br> Then <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-7-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-215\" style=\"width: 6.06em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.221em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.08em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-216\"><span class=\"mrow\" id=\"MathJax-Span-217\"><span class=\"mo\" id=\"MathJax-Span-218\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-219\"><span style=\"display: inline-block; position: relative; width: 4.289em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-220\"><span class=\"mrow\" id=\"MathJax-Span-221\"><span class=\"mn\" id=\"MathJax-Span-222\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-229\"><span class=\"mrow\" id=\"MathJax-Span-230\"><span class=\"mn\" id=\"MathJax-Span-231\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-238\"><span class=\"mrow\" id=\"MathJax-Span-239\"><span class=\"mo\" id=\"MathJax-Span-240\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-241\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.51em, 6.06em, -999.998em); top: -4.471em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-223\"><span class=\"mrow\" id=\"MathJax-Span-224\"><span class=\"mn\" id=\"MathJax-Span-225\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.336em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-232\"><span class=\"mrow\" id=\"MathJax-Span-233\"><span class=\"mi\" id=\"MathJax-Span-234\" style=\"font-family: MathJax_Math-italic;\">p<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-242\"><span class=\"mrow\" id=\"MathJax-Span-243\"><span class=\"mn\" id=\"MathJax-Span-244\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-226\"><span class=\"mrow\" id=\"MathJax-Span-227\"><span class=\"mn\" id=\"MathJax-Span-228\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-235\"><span class=\"mrow\" id=\"MathJax-Span-236\"><span class=\"mn\" id=\"MathJax-Span-237\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-245\"><span class=\"mrow\" id=\"MathJax-Span-246\"><span class=\"mn\" id=\"MathJax-Span-247\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-248\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-7\">\\left|\\begin{array}{ccc}  1 & 1 & 1 \\\\  1 & p & 2 \\\\  -1 & 1 & 1  \\end{array}\\right|<\/script>[a\u0305 b\u0305 c\u0305] = 0<br> \u21d2 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-8-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-249\" style=\"width: 6.06em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.221em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.08em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-250\"><span class=\"mrow\" id=\"MathJax-Span-251\"><span class=\"mo\" id=\"MathJax-Span-252\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-253\"><span style=\"display: inline-block; position: relative; width: 4.289em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-254\"><span class=\"mrow\" id=\"MathJax-Span-255\"><span class=\"mn\" id=\"MathJax-Span-256\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-263\"><span class=\"mrow\" id=\"MathJax-Span-264\"><span class=\"mn\" id=\"MathJax-Span-265\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-272\"><span class=\"mrow\" id=\"MathJax-Span-273\"><span class=\"mo\" id=\"MathJax-Span-274\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-275\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.51em, 6.06em, -999.998em); top: -4.471em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-257\"><span class=\"mrow\" id=\"MathJax-Span-258\"><span class=\"mn\" id=\"MathJax-Span-259\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.336em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-266\"><span class=\"mrow\" id=\"MathJax-Span-267\"><span class=\"mi\" id=\"MathJax-Span-268\" style=\"font-family: MathJax_Math-italic;\">p<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-276\"><span class=\"mrow\" id=\"MathJax-Span-277\"><span class=\"mn\" id=\"MathJax-Span-278\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-260\"><span class=\"mrow\" id=\"MathJax-Span-261\"><span class=\"mn\" id=\"MathJax-Span-262\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-269\"><span class=\"mrow\" id=\"MathJax-Span-270\"><span class=\"mn\" id=\"MathJax-Span-271\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-279\"><span class=\"mrow\" id=\"MathJax-Span-280\"><span class=\"mn\" id=\"MathJax-Span-281\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-282\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-8\">\\left|\\begin{array}{rrr}  1 & 1 & 1 \\\\  1 & p & 2 \\\\  -1 & 1 & 1  \\end{array}\\right|<\/script> = 0 (\u2235 [a\u0305 b\u0305 c\u0305] = 0)<br> \u21d2 1 (p \u2013 2) \u2013 1 (1 + 2) + 1 (1 + p) = 0<br> \u21d2 2p = 4 \u21d2 p = 2<\/p>\n\n<h3>Question 8.<br> Determine \u03bb for which the volume of the parallelopiped having coterminus edges i\u0305 + j\u0305, 3i\u0305 \u2013 j\u0305 and 3j\u0305 + \u03bb.k\u0305 is 16 cubic units.<\/h3>\n\n<p>Answer:<br> Denoting the coterminus edges by a\u0305,b\u0305,c\u0305 the volume of the parallelopiped =<br> |[a\u0305 b\u0305 c\u0305]| = \u00b116<br> \u2234 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-9-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-283\" style=\"width: 6.2em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.315em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.17em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-284\"><span class=\"mrow\" id=\"MathJax-Span-285\"><span class=\"mo\" id=\"MathJax-Span-286\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-287\"><span style=\"display: inline-block; position: relative; width: 4.383em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-288\"><span class=\"mrow\" id=\"MathJax-Span-289\"><span class=\"mn\" id=\"MathJax-Span-290\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-297\"><span class=\"mrow\" id=\"MathJax-Span-298\"><span class=\"mn\" id=\"MathJax-Span-299\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-307\"><span class=\"mrow\" id=\"MathJax-Span-308\"><span class=\"mn\" id=\"MathJax-Span-309\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.107em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-291\"><span class=\"mrow\" id=\"MathJax-Span-292\"><span class=\"mn\" id=\"MathJax-Span-293\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-300\"><span class=\"mrow\" id=\"MathJax-Span-301\"><span class=\"mo\" id=\"MathJax-Span-302\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-303\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-310\"><span class=\"mrow\" id=\"MathJax-Span-311\"><span class=\"mn\" id=\"MathJax-Span-312\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.56em, 6.06em, -999.998em); top: -4.471em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 0.608em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-294\"><span class=\"mrow\" id=\"MathJax-Span-295\"><span class=\"mn\" id=\"MathJax-Span-296\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-304\"><span class=\"mrow\" id=\"MathJax-Span-305\"><span class=\"mn\" id=\"MathJax-Span-306\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.56em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-313\"><span class=\"mrow\" id=\"MathJax-Span-314\"><span class=\"mi\" id=\"MathJax-Span-315\" style=\"font-family: MathJax_Math-italic;\">\u03bb<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-316\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-9\">\\left|\\begin{array}{rrr}  1 & 1 & 0 \\\\  3 & -1 & 0 \\\\  0 & 3 & \\lambda  \\end{array}\\right|<\/script> = \u00b116<br> \u21d2 1(-\u03bb) \u2013 1(3\u03bb) = \u00b116<br> \u21d2 \u2013 4\u03bb = \u00b116<br> \u21d2 \u03bb = \u00b14<\/p>\n\n\n\n<h3>Question 9.<br> Find the volume of the tetrahedron having the edges i\u0305 + j\u0305 + k\u0305; i\u0305 \u2013 j\u0305 and i\u0305 + 2j\u0305 + k\u0305.<\/h3>\n\n<p>Answer:<br> Denoting the edges by a\u0305, b\u0305, c\u0305 of tetrahedron, then its volume is = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-10-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-317\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-318\"><span class=\"mfrac\" id=\"MathJax-Span-319\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-320\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-321\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">6<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-10\">\\frac{1}{6}<\/script>[a\u0305 b\u0305 c\u0305]<br>\n\n<img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-7242\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5c-1.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 1\" width=\"258\" height=\"199\" data-pin-nopin=\"true\"><\/p>\n\n<h3>Question 10.<br> Let a\u0305, b\u0305 and c\u0305 be non coplanar vectors and \u03b1 = a\u0305 + 2b\u0305 + 3c\u0305, \u03b2 = 2a\u0305 + b\u0305 \u2013 2c\u0305 and \u03b3 = 3a\u0305 \u2013 7c\u0305, then find [\u03b1\u0305 \u03b2\u0305 \u03b3\u0305].<\/h3>\n\n<p>Answer:<br> Given \u03b1 = a\u0305 + 2b\u0305 + 3c\u0305<br> \u03b2 = 2a\u0305 + b\u0305 \u2013 2c\u0305<br> \u03b3 = 3a\u0305 \u2013 7c\u0305<br> and a\u0305, b\u0305, c\u0305 are non coplanar \u21d2 [a\u0305 b\u0305 c\u0305] \u2260 0<br> then [\u03b1\u0305 \u03b2\u0305 \u03b3\u0305] = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-11-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-322\" style=\"width: 6.06em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.221em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.08em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-323\"><span class=\"mrow\" id=\"MathJax-Span-324\"><span class=\"mo\" id=\"MathJax-Span-325\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-326\"><span style=\"display: inline-block; position: relative; width: 4.289em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-327\"><span class=\"mrow\" id=\"MathJax-Span-328\"><span class=\"mn\" id=\"MathJax-Span-329\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-336\"><span class=\"mrow\" id=\"MathJax-Span-337\"><span class=\"mn\" id=\"MathJax-Span-338\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-346\"><span class=\"mrow\" id=\"MathJax-Span-347\"><span class=\"mn\" id=\"MathJax-Span-348\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-330\"><span class=\"mrow\" id=\"MathJax-Span-331\"><span class=\"mn\" id=\"MathJax-Span-332\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-339\"><span class=\"mrow\" id=\"MathJax-Span-340\"><span class=\"mn\" id=\"MathJax-Span-341\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-349\"><span class=\"mrow\" id=\"MathJax-Span-350\"><span class=\"mn\" id=\"MathJax-Span-351\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.26em, 6.247em, -999.998em); top: -4.564em; left: 2.985em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-333\"><span class=\"mrow\" id=\"MathJax-Span-334\"><span class=\"mn\" id=\"MathJax-Span-335\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-342\"><span class=\"mrow\" id=\"MathJax-Span-343\"><span class=\"mo\" id=\"MathJax-Span-344\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-345\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1001.26em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-352\"><span class=\"mrow\" id=\"MathJax-Span-353\"><span class=\"mo\" id=\"MathJax-Span-354\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-355\" style=\"font-family: MathJax_Main;\">7<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-356\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-11\">\\left|\\begin{array}{rrr}  1 & 2 & 3 \\\\  2 & 1 & -2 \\\\  3 & 0 & -7  \\end{array}\\right|<\/script>[a\u0305 b\u0305 c\u0305]<br> = [ 1 (- 7) \u2013 2 (- 14 + 6) + 3 (- 3)] [a\u0305 b\u0305 c\u0305]<br> = (- 7 + 16 \u2013 9) [a\u0305 b\u0305 c\u0305] = 0<\/p>\n\n<h3>Question 11.<br> Let a\u0305, b\u0305 and c\u0305 be non coplanar vectors. If [2a\u0305 \u2013 b\u0305 + 3c\u0305, a\u0305 + b\u0305 \u2013 2c\u0305, a\u0305 + b\u0305 \u2013 3c\u0305] = \u03bb [a\u0305 b\u0305 c\u0305] then find the value of \u03bb.<\/h3>\n\n<p>Answer:<br> Given a,b,c as non coplanar vectors We have [a\u0305 b\u0305 c\u0305] \u2260 0<br> \u2234 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-12-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-357\" style=\"width: 6.992em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.014em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.87em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-358\"><span class=\"mrow\" id=\"MathJax-Span-359\"><span class=\"mo\" id=\"MathJax-Span-360\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-361\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-362\"><span class=\"mrow\" id=\"MathJax-Span-363\"><span class=\"mn\" id=\"MathJax-Span-364\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-372\"><span class=\"mrow\" id=\"MathJax-Span-373\"><span class=\"mn\" id=\"MathJax-Span-374\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-382\"><span class=\"mrow\" id=\"MathJax-Span-383\"><span class=\"mn\" id=\"MathJax-Span-384\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.06em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-365\"><span class=\"mrow\" id=\"MathJax-Span-366\"><span class=\"mo\" id=\"MathJax-Span-367\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-368\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-375\"><span class=\"mrow\" id=\"MathJax-Span-376\"><span class=\"mn\" id=\"MathJax-Span-377\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-385\"><span class=\"mrow\" id=\"MathJax-Span-386\"><span class=\"mn\" id=\"MathJax-Span-387\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.26em, 6.247em, -999.998em); top: -4.564em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-369\"><span class=\"mrow\" id=\"MathJax-Span-370\"><span class=\"mn\" id=\"MathJax-Span-371\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-378\"><span class=\"mrow\" id=\"MathJax-Span-379\"><span class=\"mo\" id=\"MathJax-Span-380\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-381\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-388\"><span class=\"mrow\" id=\"MathJax-Span-389\"><span class=\"mo\" id=\"MathJax-Span-390\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-391\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-392\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-12\">\\left|\\begin{array}{ccc}  2 & -1 & 3 \\\\  1 & 1 & -2 \\\\  1 & 1 & -3  \\end{array}\\right|<\/script>[a\u0305 b\u0305 c\u0305]<br> = [2 (- 3 + 2) + 1 (- 3 + 2) + 3 (1 \u2013 1)][a\u0305 b\u0305 c\u0305]<br> = [-2 \u2013 1] [a\u0305 b\u0305 c\u0305]<br> = -3[a\u0305 b\u0305 c\u0305]<br> Given [2a\u0305 \u2013 b\u0305 + 3c\u0305, a\u0305 + b\u0305 \u2013 2c\u0305, a\u0305 + b\u0305 \u2013 3c\u0305]<br> = \u03bb [a b c]<br> We have -3[a\u0305 b\u0305 c\u0305] = \u03bb [a\u0305 b\u0305 c\u0305]<br> \u21d2 \u03bb = -3<\/p>\n\n\n\n<h3>Question 12.<br> Let a\u0305, b\u0305 and c\u0305 be non coplanar vectors.<br> If [a\u0305 + 2b\u0305 2b\u0305 + c\u0305 5c\u0305 + a\u0305] = \u03bb [a\u0305 b\u0305 c\u0305], then find \u03bb.<\/h3>\n\n<p>Answer:<br> Given a\u0305, b\u0305 and c\u0305 as non coplanar vectors. We have [a\u0305 b\u0305 c\u0305] \u2260 0. Given that<br> \u2234 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-13-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-393\" style=\"width: 5.128em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.429em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1004.29em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-394\"><span class=\"mrow\" id=\"MathJax-Span-395\"><span class=\"mo\" id=\"MathJax-Span-396\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-397\"><span style=\"display: inline-block; position: relative; width: 3.497em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-398\"><span class=\"mrow\" id=\"MathJax-Span-399\"><span class=\"mn\" id=\"MathJax-Span-400\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-407\"><span class=\"mrow\" id=\"MathJax-Span-408\"><span class=\"mn\" id=\"MathJax-Span-409\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-416\"><span class=\"mrow\" id=\"MathJax-Span-417\"><span class=\"mn\" id=\"MathJax-Span-418\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-401\"><span class=\"mrow\" id=\"MathJax-Span-402\"><span class=\"mn\" id=\"MathJax-Span-403\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-410\"><span class=\"mrow\" id=\"MathJax-Span-411\"><span class=\"mn\" id=\"MathJax-Span-412\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-419\"><span class=\"mrow\" id=\"MathJax-Span-420\"><span class=\"mn\" id=\"MathJax-Span-421\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 2.985em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-404\"><span class=\"mrow\" id=\"MathJax-Span-405\"><span class=\"mn\" id=\"MathJax-Span-406\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-413\"><span class=\"mrow\" id=\"MathJax-Span-414\"><span class=\"mn\" id=\"MathJax-Span-415\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-422\"><span class=\"mrow\" id=\"MathJax-Span-423\"><span class=\"mn\" id=\"MathJax-Span-424\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-425\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-13\">\\left|\\begin{array}{lll}  1 & 2 & 0 \\\\  0 & 2 & 1 \\\\  1 & 0 & 5  \\end{array}\\right|<\/script>[a\u0305 b\u0305 c\u0305] = \u03bb[a\u0305 b\u0305 c\u0305]<br> \u21d2 [1 (10 \u2013 0) \u2013 2 (0 \u2013 1)] [a\u0305 b\u0305 c\u0305] = \u03bb[a\u0305 b\u0305 c\u0305]<br> \u21d2 12 [a\u0305 b\u0305 c\u0305] = \u03bb [a\u0305 b\u0305 c\u0305]<br> \u21d2 \u03bb = 12<\/p>\n\n<h3>Question 13.<br> If a\u0305, b\u0305, c\u0305 are non coplanar vectors, then find the value of <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-14-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-426\" style=\"width: 10.673em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 9.182em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(0.841em, 1009.18em, 3.218em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-427\"><span class=\"mfrac\" id=\"MathJax-Span-428\"><span style=\"display: inline-block; position: relative; width: 8.903em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.124em, 1008.67em, 4.336em, -999.998em); top: -4.564em; left: 50%; margin-left: -4.378em;\"><span class=\"mrow\" id=\"MathJax-Span-429\"><span class=\"mo\" id=\"MathJax-Span-430\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">(<\/span><span class=\"munderover\" id=\"MathJax-Span-431\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-432\"><span class=\"mrow\" id=\"MathJax-Span-433\"><span class=\"mi\" id=\"MathJax-Span-434\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.497em, 1000.33em, 3.87em, -999.998em); top: -4.192em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-435\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.329em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.049em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-436\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-437\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"munderover\" id=\"MathJax-Span-438\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.38em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-439\"><span class=\"mrow\" id=\"MathJax-Span-440\"><span class=\"mi\" id=\"MathJax-Span-441\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.497em, 1000.38em, 3.87em, -999.998em); top: -4.378em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-442\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.189em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-443\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"munderover\" id=\"MathJax-Span-444\"><span style=\"display: inline-block; position: relative; width: 0.329em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.28em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-445\"><span class=\"mrow\" id=\"MathJax-Span-446\"><span class=\"mi\" id=\"MathJax-Span-447\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.497em, 1000.33em, 3.87em, -999.998em); top: -4.192em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-448\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.329em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.049em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-449\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">)<\/span><span class=\"mo\" id=\"MathJax-Span-450\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u22c5<\/span><span class=\"mo\" id=\"MathJax-Span-451\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">[<\/span><span class=\"mo\" id=\"MathJax-Span-452\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">(<\/span><span class=\"munderover\" id=\"MathJax-Span-453\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-454\"><span class=\"mrow\" id=\"MathJax-Span-455\"><span class=\"mi\" id=\"MathJax-Span-456\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.497em, 1000.33em, 3.87em, -999.998em); top: -4.192em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-457\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.329em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.049em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-458\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"munderover\" id=\"MathJax-Span-459\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.38em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-460\"><span class=\"mrow\" id=\"MathJax-Span-461\"><span class=\"mi\" id=\"MathJax-Span-462\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.497em, 1000.38em, 3.87em, -999.998em); top: -4.378em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-463\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.189em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-464\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">)<\/span><span class=\"mo\" id=\"MathJax-Span-465\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00d7<\/span><span class=\"mo\" id=\"MathJax-Span-466\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">(<\/span><span class=\"munderover\" id=\"MathJax-Span-467\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-468\"><span class=\"mrow\" id=\"MathJax-Span-469\"><span class=\"mi\" id=\"MathJax-Span-470\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.497em, 1000.33em, 3.87em, -999.998em); top: -4.192em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-471\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.329em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.049em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-472\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"munderover\" id=\"MathJax-Span-473\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.38em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-474\"><span class=\"mrow\" id=\"MathJax-Span-475\"><span class=\"mi\" id=\"MathJax-Span-476\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.497em, 1000.38em, 3.87em, -999.998em); top: -4.378em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-477\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.189em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-478\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"munderover\" id=\"MathJax-Span-479\"><span style=\"display: inline-block; position: relative; width: 0.329em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.28em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-480\"><span class=\"mrow\" id=\"MathJax-Span-481\"><span class=\"mi\" id=\"MathJax-Span-482\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.497em, 1000.33em, 3.87em, -999.998em); top: -4.192em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-483\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.329em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.049em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-484\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">)<\/span><span class=\"mo\" id=\"MathJax-Span-485\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">]<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.124em, 1001.35em, 4.336em, -999.998em); top: -3.399em; left: 50%; margin-left: -0.743em;\"><span class=\"mrow\" id=\"MathJax-Span-486\"><span class=\"mo\" id=\"MathJax-Span-487\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">[<\/span><span class=\"munderover\" id=\"MathJax-Span-488\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-489\"><span class=\"mrow\" id=\"MathJax-Span-490\"><span class=\"mi\" id=\"MathJax-Span-491\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.497em, 1000.33em, 3.87em, -999.998em); top: -4.192em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-492\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.329em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.049em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"munderover\" id=\"MathJax-Span-493\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.38em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-494\"><span class=\"mrow\" id=\"MathJax-Span-495\"><span class=\"mi\" id=\"MathJax-Span-496\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.497em, 1000.38em, 3.87em, -999.998em); top: -4.378em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-497\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.189em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"munderover\" id=\"MathJax-Span-498\"><span style=\"display: inline-block; position: relative; width: 0.329em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.28em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-499\"><span class=\"mrow\" id=\"MathJax-Span-500\"><span class=\"mi\" id=\"MathJax-Span-501\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.497em, 1000.33em, 3.87em, -999.998em); top: -4.192em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-502\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.329em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.049em;\"><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-503\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">]<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1008.9em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 8.903em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.97em; border-left: 0px solid; width: 0px; height: 2.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-14\">\\frac{(\\overline{\\mathrm{a}}+2 \\overline{\\mathrm{b}}-\\overline{\\mathrm{c}}) \\cdot[(\\overline{\\mathrm{a}}-\\overline{\\mathrm{b}}) \\times(\\overline{\\mathrm{a}}-\\overline{\\mathrm{b}}-\\overline{\\mathrm{c}})]}{[\\overline{\\mathrm{a}} \\overline{\\mathrm{b}} \\overline{\\mathrm{c}}]}<\/script><br><\/h3>\n\n<p>Answer:<br> Given a\u0305, b\u0305, c\u0305 are non coplanar we have [a\u0305 b\u0305 c\u0305] \u2260 0, then<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7241\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5c-2.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 2\" width=\"285\" height=\"162\" data-pin-nopin=\"true\"><br> = 1 (1) \u2013 2 (- 1) \u2013 1 (- 1 + 1) = 3<\/p>\n\n<h3>Question 14.<br> If a\u0305, b\u0305, c\u0305 are mutually perpendicular unit vectors, then find the value of [a\u0305 b\u0305 c\u0305]<sup>2<\/sup>.<\/h3>\n\n<p>Answer:<br> Given a\u0305, b\u0305, c\u0305 are mutually perpendicular unit vectors.<br> We have |a\u0305| = |b\u0305| = |c\u0305| = 1<br> and taking a\u0305 = i\u0305, b\u0305 = j\u0305, c\u0305 = k\u0305<br> We have [a\u0305 b\u0305 c\u0305] = [i\u0305 j\u0305 k\u0305]<br> = i\u0305 . (j\u0305 \u00d7 k\u0305) = i\u0305.i\u0305 = 1<br> \u2234 [a\u0305 b\u0305 c\u0305]<sup>2<\/sup> = 1<\/p>\n\n\n\n<h3>Question 15.<br> a\u0305, b\u0305, c\u0305 are non zero vectors and a\u0305 is perpendicular to both b\u0305 and c\u0305. If |a\u0305|= 2, |b\u0305|= 3, |c\u0305| = 4 and (b\u0305, c\u0305) = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-15-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-504\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-505\"><span class=\"mfrac\" id=\"MathJax-Span-506\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.75em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-507\"><span class=\"mn\" id=\"MathJax-Span-508\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-509\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-510\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-15\">\\frac{2 \\pi}{3}<\/script> then find |[a\u0305 b\u0305 c\u0305]|. (May 2008)<\/h3>\n\n<p>Answer:<br> Given a\u0305, b\u0305, c\u0305 are non zero vectors and a is perpendicular to both b\u0305 and c\u0305<br> \u21d2 a\u0305 is parallel to (b\u0305 \u00d7 c\u0305)<br> \u21d2 (a\u0305, b\u0305 \u00d7 c\u0305) = 0 (or) 180\u00b0<br> \u2234 [a\u0305 b\u0305 c\u0305] = [a\u0305 . b\u0305 \u00d7 c\u0305]<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7240\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5c-3.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 3\" width=\"322\" height=\"251\" sizes=\"auto, (max-width: 322px) 100vw, 322px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 3\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c)\"><\/p>\n\n<h3>Question 16.<br> If a\u0305, b\u0305, c\u0305 are unit coplanar vectors, then find [2a\u0305 \u2013 b\u0305 2b\u0305 \u2013 c\u0305 2c\u0305 \u2013 a\u0305].<\/h3>\n\n<p>Answer:<br> Given |a\u0305| = |b\u0305| = |c\u0305| = 1<br> [2a\u0305 \u2013 b\u0305 2b\u0305 \u2013 c\u0305 2c\u0305 \u2013 a\u0305]<br> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-16-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-511\" style=\"width: 7.831em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.759em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1006.62em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-512\"><span class=\"mrow\" id=\"MathJax-Span-513\"><span class=\"mo\" id=\"MathJax-Span-514\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-515\"><span style=\"display: inline-block; position: relative; width: 5.827em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-516\"><span class=\"mrow\" id=\"MathJax-Span-517\"><span class=\"mn\" id=\"MathJax-Span-518\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-526\"><span class=\"mrow\" id=\"MathJax-Span-527\"><span class=\"mn\" id=\"MathJax-Span-528\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-536\"><span class=\"mrow\" id=\"MathJax-Span-537\"><span class=\"mo\" id=\"MathJax-Span-538\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-539\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.107em, -999.998em); top: -4.471em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-519\"><span class=\"mrow\" id=\"MathJax-Span-520\"><span class=\"mo\" id=\"MathJax-Span-521\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-522\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-529\"><span class=\"mrow\" id=\"MathJax-Span-530\"><span class=\"mn\" id=\"MathJax-Span-531\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-540\"><span class=\"mrow\" id=\"MathJax-Span-541\"><span class=\"mn\" id=\"MathJax-Span-542\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.06em, -999.998em); top: -4.471em; left: 4.569em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-523\"><span class=\"mrow\" id=\"MathJax-Span-524\"><span class=\"mn\" id=\"MathJax-Span-525\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-532\"><span class=\"mrow\" id=\"MathJax-Span-533\"><span class=\"mo\" id=\"MathJax-Span-534\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-535\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-543\"><span class=\"mrow\" id=\"MathJax-Span-544\"><span class=\"mn\" id=\"MathJax-Span-545\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-546\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-16\">\\left|\\begin{array}{rrr}  2 & -1 & 0 \\\\  0 & 2 & -1 \\\\  -1 & 0 & 2  \\end{array}\\right|<\/script>[a\u0305 b\u0305 c\u0305]<br> = [2 (4) + 1 (- 1)] [a\u0305 b\u0305 c\u0305]<br> = 7 [a\u0305 b\u0305 c\u0305] = 7 (0) = 0 (\u2235 a\u0305, b\u0305, c\u0305 are coplanar vectors [a\u0305 b\u0305 c\u0305] = 0 ]<\/p>\n\n<h3>II.<br> Question 1.<br> If [b\u0305 c\u0305 d\u0305] + [c\u0305 a\u0305 d\u0305] + [a\u0305 b\u0305 d\u0305] = [a\u0305 b\u0305 c\u0305], then show that the points with position vectors a\u0305, b\u0305, c\u0305 and d\u0305 are coplanar. (May 2014)<\/h3>\n\n<p>Answer:<br> Given [b\u0305 c\u0305 d\u0305] + [c\u0305 a\u0305 d\u0305] + [a\u0305 b\u0305 d\u0305]<br> = [a\u0305 b\u0305 c\u0305] \u2026\u2026\u2026..(1)<br> Let OA = a\u0305, OB = b\u0305, OC = c\u0305 and OD = d\u0305 with respect to a fixed origin \u2018O\u2019. Then<br> AB = b\u0305 \u2013 a\u0305,<br> AC = c\u0305 \u2013 a\u0305,<br> AD = d\u0305 \u2013 a\u0305 if the points A, B, C, D are coplanar then<br> [AB AC AD] = 0<br> \u21d2 [b\u0305 \u2013 a\u0305 c\u0305 \u2013 a\u0305 d\u0305 \u2013 a\u0305] = 0<br> \u21d2 (b\u0305 \u2013 a\u0305) . [(c\u0305 \u2013 a\u0305) \u00d7 (d\u0305\u0305 \u2013 a\u0305)] = 0<br> \u21d2 (b\u0305 \u2013 a\u0305) . [c\u0305 \u00d7 d\u0305 \u2013 (c\u0305 \u00d7 a\u0305) \u2013 (a\u0305 \u00d7 d\u0305) + (a\u0305 \u00d7 a\u0305)] = 0<br> \u21d2 (b\u0305 \u2013 a\u0305) \u2013 [(c\u0305 \u00d7 d\u0305) \u2013 (a\u0305 \u00d7 d\u0305) \u2013 (c\u0305 \u00d7 a\u0305)] = 0<br> \u21d2 [b\u0305 c\u0305 d\u0305] \u2013 (b\u0305 a\u0305 d\u0305) \u2013 [b\u0305 c\u0305 a\u0305] \u2013 [a\u0305 c\u0305 d\u0305] + [a\u0305 a\u0305 d\u0305] + [a\u0305 c\u0305 a\u0305] = 0<br> \u21d2 [b\u0305 c\u0305 d\u0305] \u2013 [b\u0305 a\u0305 d\u0305] \u2013 [b\u0305 c\u0305 a\u0305] \u2013 [a\u0305 c\u0305 d\u0305] = 0<br> \u21d2 [b\u0305 c\u0305 d\u0305] + [a\u0305 b\u0305 a\u0305] \u2013 [a\u0305 b\u0305 c\u0305] + [c\u0305 a\u0305 d\u0305] = 0<br> \u21d2 [b\u0305 c\u0305 d\u0305] + [a\u0305 b\u0305 d\u0305] + [c\u0305 a\u0305 d\u0305] = [a\u0305 b\u0305 c\u0305]<\/p>\n\n<h3>Question 2.<br> If a\u0305, b\u0305 and c\u0305 are non coplanar vectors, then prove that the four points with position vectors 2a\u0305 + 3b\u0305 \u2013 c\u0305, a\u0305 \u2013 2b\u0305 + 3c\u0305, 3a\u0305 + 4b\u0305 \u2013 2c\u0305 and a\u0305 \u2013 6b\u0305 + 6c\u0305 are coplanar.<\/h3>\n\n<p>Answer:<br> Suppose A, B, C, D are the given points with respect to a fixed origin \u2018O\u2019 and given that<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-17-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-547\" style=\"width: 1.82em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.54em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.074em, 1001.54em, 2.425em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-548\"><span class=\"munderover\" id=\"MathJax-Span-549\"><span style=\"display: inline-block; position: relative; width: 1.54em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.49em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-550\"><span class=\"mrow\" id=\"MathJax-Span-551\"><span class=\"mi\" id=\"MathJax-Span-552\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-553\" style=\"font-family: MathJax_Main;\">A<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.49em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-554\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.214em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.515em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.701em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.888em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.074em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.051em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-17\">\\overline{\\mathrm{OA}}<\/script> = 2a\u0305 + 3b\u0305 \u2013 c\u0305, <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-18-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-555\" style=\"width: 1.726em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1001.49em, 2.425em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-556\"><span class=\"munderover\" id=\"MathJax-Span-557\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.45em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-558\"><span class=\"mrow\" id=\"MathJax-Span-559\"><span class=\"mi\" id=\"MathJax-Span-560\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-561\" style=\"font-family: MathJax_Main;\">B<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-562\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.375em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.795em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.051em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-18\">\\overline{\\mathrm{OB}}<\/script> = a\u0305 \u2013 2b\u0305 + 3c\u0305<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-19-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-563\" style=\"width: 1.726em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1001.49em, 2.425em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-564\"><span class=\"munderover\" id=\"MathJax-Span-565\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.45em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-566\"><span class=\"mrow\" id=\"MathJax-Span-567\"><span class=\"mi\" id=\"MathJax-Span-568\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-569\" style=\"font-family: MathJax_Main;\">C<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.49em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-570\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.282em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.468em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.655em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.841em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.027em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.051em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-19\">\\overline{\\mathrm{OC}}<\/script> = 3a\u0305 + 4b\u0305 \u2013 2c\u0305 and <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-20-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-571\" style=\"width: 1.82em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.54em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1001.54em, 2.425em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-572\"><span class=\"munderover\" id=\"MathJax-Span-573\"><span style=\"display: inline-block; position: relative; width: 1.54em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.49em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-574\"><span class=\"mrow\" id=\"MathJax-Span-575\"><span class=\"mi\" id=\"MathJax-Span-576\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-577\" style=\"font-family: MathJax_Main;\">D<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.54em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-578\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.54em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.214em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.515em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.701em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.888em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.074em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.051em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-20\">\\overline{\\mathrm{OD}}<\/script> = a\u0305 \u2013 6b\u0305 + 6c\u0305<br> \u2234 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-21-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-579\" style=\"width: 8.25em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 7.085em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1007.09em, 2.565em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-580\"><span class=\"munderover\" id=\"MathJax-Span-581\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.4em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-582\"><span class=\"mrow\" id=\"MathJax-Span-583\"><span class=\"mi\" id=\"MathJax-Span-584\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-585\" style=\"font-family: MathJax_Main;\">B<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-586\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.121em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.748em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-587\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-588\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.45em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-589\"><span class=\"mrow\" id=\"MathJax-Span-590\"><span class=\"mi\" id=\"MathJax-Span-591\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-592\" style=\"font-family: MathJax_Main;\">B<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-593\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.375em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.795em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-594\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"munderover\" id=\"MathJax-Span-595\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 1.54em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.49em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-596\"><span class=\"mrow\" id=\"MathJax-Span-597\"><span class=\"mi\" id=\"MathJax-Span-598\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-599\" style=\"font-family: MathJax_Main;\">A<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.49em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-600\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.214em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.515em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.701em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.888em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.074em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.159em; border-left: 0px solid; width: 0px; height: 1.408em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-21\">\\overline{\\mathrm{AB}}=\\overline{\\mathrm{OB}}-\\overline{\\mathrm{OA}}<\/script><br> = (a\u0305 \u2013 2b\u0305 + 3c\u0305) \u2013 (2a\u0305 + 3b\u0305 \u2013 c\u0305)<br> = -a\u0305 \u2013 5b\u0305 + 4c\u0305<\/p>\n\n <p><span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-22-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-601\" style=\"width: 8.297em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 7.132em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1007.13em, 2.565em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-602\"><span class=\"munderover\" id=\"MathJax-Span-603\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.4em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-604\"><span class=\"mrow\" id=\"MathJax-Span-605\"><span class=\"mi\" id=\"MathJax-Span-606\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-607\" style=\"font-family: MathJax_Main;\">C<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-608\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.795em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-609\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-610\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.45em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-611\"><span class=\"mrow\" id=\"MathJax-Span-612\"><span class=\"mi\" id=\"MathJax-Span-613\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-614\" style=\"font-family: MathJax_Main;\">C<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.49em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-615\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.282em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.468em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.655em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.841em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.027em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-616\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"munderover\" id=\"MathJax-Span-617\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 1.54em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.49em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-618\"><span class=\"mrow\" id=\"MathJax-Span-619\"><span class=\"mi\" id=\"MathJax-Span-620\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-621\" style=\"font-family: MathJax_Main;\">A<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.49em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-622\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.214em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.515em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.701em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.888em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.074em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.159em; border-left: 0px solid; width: 0px; height: 1.408em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-22\">\\overline{\\mathrm{AC}}=\\overline{\\mathrm{OC}}-\\overline{\\mathrm{OA}}<\/script><br> = (3a\u0305 + 4b\u0305 \u2013 2c\u0305) \u2013 (2a\u0305 + 3b\u0305 \u2013 c\u0305)<br> = a\u0305 + b\u0305 \u2013 c\u0305<\/p>\n\n<p><span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-23-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-623\" style=\"width: 8.343em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 7.178em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1007.18em, 2.565em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-624\"><span class=\"munderover\" id=\"MathJax-Span-625\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.45em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-626\"><span class=\"mrow\" id=\"MathJax-Span-627\"><span class=\"mi\" id=\"MathJax-Span-628\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-629\" style=\"font-family: MathJax_Main;\">D<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.49em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-630\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.282em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.468em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.655em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.841em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.027em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-631\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-632\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.54em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.49em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-633\"><span class=\"mrow\" id=\"MathJax-Span-634\"><span class=\"mi\" id=\"MathJax-Span-635\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-636\" style=\"font-family: MathJax_Main;\">D<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.54em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-637\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.54em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.214em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.515em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.701em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.888em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.074em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-638\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"munderover\" id=\"MathJax-Span-639\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 1.54em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.49em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-640\"><span class=\"mrow\" id=\"MathJax-Span-641\"><span class=\"mi\" id=\"MathJax-Span-642\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-643\" style=\"font-family: MathJax_Main;\">A<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.49em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-644\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.214em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.515em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.701em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.888em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.074em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.159em; border-left: 0px solid; width: 0px; height: 1.408em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-23\">\\overline{\\mathrm{AD}}=\\overline{\\mathrm{OD}}-\\overline{\\mathrm{OA}}<\/script><br> = (a\u0305 \u2013 6b\u0305 + 6c\u0305) \u2013 (2a\u0305 + 3b\u0305 \u2013 c\u0305)<br> = -a\u0305 \u2013 9b\u0305 + 7c\u0305<\/p>\n\n\n\n<p>\u2234 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-24-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-645\" style=\"width: 24.047em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 20.739em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.286em, 1020.55em, 6.34em, -999.998em); top: -4.564em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-646\"><span class=\"mrow\" id=\"MathJax-Span-647\"><span class=\"mo\" id=\"MathJax-Span-648\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-649\"><span style=\"display: inline-block; position: relative; width: 6.433em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.985em, 1001.45em, 4.336em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.798em, 1001.45em, 4.15em, -999.998em); top: -3.819em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-650\"><span class=\"mrow\" id=\"MathJax-Span-651\"><span class=\"munderover\" id=\"MathJax-Span-652\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.4em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-653\"><span class=\"mrow\" id=\"MathJax-Span-654\"><span class=\"mi\" id=\"MathJax-Span-655\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-656\" style=\"font-family: MathJax_Main;\">B<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-657\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.121em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.748em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1001.45em, 4.336em, -999.998em); top: -4.005em; left: 2.472em;\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.798em, 1001.45em, 4.15em, -999.998em); top: -3.819em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-658\"><span class=\"mrow\" id=\"MathJax-Span-659\"><span class=\"munderover\" id=\"MathJax-Span-660\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.4em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-661\"><span class=\"mrow\" id=\"MathJax-Span-662\"><span class=\"mi\" id=\"MathJax-Span-663\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-664\" style=\"font-family: MathJax_Main;\">C<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-665\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.795em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1001.49em, 4.336em, -999.998em); top: -4.005em; left: 4.942em;\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.798em, 1001.49em, 4.15em, -999.998em); top: -3.819em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-666\"><span class=\"mrow\" id=\"MathJax-Span-667\"><span class=\"munderover\" id=\"MathJax-Span-668\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.45em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-669\"><span class=\"mrow\" id=\"MathJax-Span-670\"><span class=\"mi\" id=\"MathJax-Span-671\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-672\" style=\"font-family: MathJax_Main;\">D<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.49em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-673\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.282em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.468em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.655em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.841em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.027em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-674\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">]<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-675\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mrow\" id=\"MathJax-Span-676\" style=\"padding-left: 0.282em;\"><span class=\"mo\" id=\"MathJax-Span-677\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-678\"><span style=\"display: inline-block; position: relative; width: 5.827em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-679\"><span class=\"mrow\" id=\"MathJax-Span-680\"><span class=\"mo\" id=\"MathJax-Span-681\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-682\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-690\"><span class=\"mrow\" id=\"MathJax-Span-691\"><span class=\"mn\" id=\"MathJax-Span-692\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-700\"><span class=\"mrow\" id=\"MathJax-Span-701\"><span class=\"mo\" id=\"MathJax-Span-702\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-703\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-683\"><span class=\"mrow\" id=\"MathJax-Span-684\"><span class=\"mo\" id=\"MathJax-Span-685\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-686\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-693\"><span class=\"mrow\" id=\"MathJax-Span-694\"><span class=\"mn\" id=\"MathJax-Span-695\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-704\"><span class=\"mrow\" id=\"MathJax-Span-705\"><span class=\"mo\" id=\"MathJax-Span-706\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-707\" style=\"font-family: MathJax_Main;\">9<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1001.21em, 6.153em, -999.998em); top: -4.518em; left: 4.569em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-687\"><span class=\"mrow\" id=\"MathJax-Span-688\"><span class=\"mn\" id=\"MathJax-Span-689\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-696\"><span class=\"mrow\" id=\"MathJax-Span-697\"><span class=\"mo\" id=\"MathJax-Span-698\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-699\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-708\"><span class=\"mrow\" id=\"MathJax-Span-709\"><span class=\"mn\" id=\"MathJax-Span-710\" style=\"font-family: MathJax_Main;\">7<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-711\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span class=\"mrow\" id=\"MathJax-Span-712\" style=\"padding-left: 0.189em;\"><span class=\"mo\" id=\"MathJax-Span-713\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-714\"><span style=\"display: inline-block; position: relative; width: 3.59em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(3.264em, 1000.56em, 4.336em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.078em, 1000.56em, 4.15em, -999.998em); top: -3.819em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-715\"><span class=\"mrow\" id=\"MathJax-Span-716\"><span class=\"munderover\" id=\"MathJax-Span-717\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.56em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-718\"><span class=\"mrow\" id=\"MathJax-Span-719\"><span class=\"mi\" id=\"MathJax-Span-720\" style=\"font-family: MathJax_Main-bold;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.51em, 3.777em, -999.998em); top: -4.285em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-721\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.235em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1000.56em, 4.336em, -999.998em); top: -4.005em; left: 1.54em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.845em, 1000.56em, 4.15em, -999.998em); top: -3.819em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-722\"><span class=\"mrow\" id=\"MathJax-Span-723\"><span class=\"munderover\" id=\"MathJax-Span-724\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-725\"><span class=\"mrow\" id=\"MathJax-Span-726\"><span class=\"mi\" id=\"MathJax-Span-727\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.51em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-728\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.235em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1000.47em, 4.336em, -999.998em); top: -4.005em; left: 3.124em;\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.078em, 1000.47em, 4.15em, -999.998em); top: -3.819em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-729\"><span class=\"mrow\" id=\"MathJax-Span-730\"><span class=\"munderover\" id=\"MathJax-Span-731\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0.002em;\"><span class=\"texatom\" id=\"MathJax-Span-732\"><span class=\"mrow\" id=\"MathJax-Span-733\"><span class=\"mi\" id=\"MathJax-Span-734\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.47em, 3.777em, -999.998em); top: -4.285em; left: 0em;\"><span class=\"mo\" id=\"MathJax-Span-735\" style=\"\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.189em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-736\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-24\">\\left[\\begin{array}{lll}  \\overline{\\mathrm{AB}} & \\overline{\\mathrm{AC}} & \\overline{\\mathrm{AD}}  \\end{array}\\right]=\\left|\\begin{array}{rrr}  -1 & -5 & 4 \\\\  1 & 1 & -1 \\\\  -1 & -9 & 7  \\end{array}\\right|\\left[\\begin{array}{lll}  \\overline{\\mathbf{a}} & \\overline{\\mathrm{b}} & \\overline{\\mathrm{c}}  \\end{array}\\right]<\/script><br> = [ \u2013 1 (7 \u2013 9) + 5 (7 \u2013 1) + 4 (- 9 + 1)] [a\u0305 b\u0305 c\u0305]<br> = [- 1 (- 2) + 5 (6) + 4 (- 8)] [a\u0305 b\u0305 c\u0305]<br> = (2 + 30 \u2013 32) [a\u0305 b\u0305 c\u0305] = 0<br> Hence the given points A, B, C, D are coplanar.<\/p>\n\n<h3>Question 3.<br> a\u0305, b\u0305 and c\u0305 are non zero and non collinear vectors and \u03b8 \u2260 0, is the angle between b\u0305 and c\u0305. If (a\u0305 \u00d7 b\u0305) \u00d7 c\u0305 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-25-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-737\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-738\"><span class=\"mfrac\" id=\"MathJax-Span-739\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-740\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-741\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-25\">\\frac{1}{3}<\/script>|b\u0305||c\u0305||a\u0305| find sin \u03b8.<\/h3>\n\n<p>Answer:<br> Given |a\u0305| \u2260 0, |b\u0305| \u2260 0, |c\u0305| \u2260 0 and (b\u0305, c\u0305) = \u03b8<br> and (a\u0305 \u00d7 b\u0305) \u00d7 c\u0305 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-26-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-742\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-743\"><span class=\"mfrac\" id=\"MathJax-Span-744\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-745\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-746\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-26\">\\frac{1}{3}<\/script>|b\u0305||c\u0305|a\u0305<br> \u21d2 (a\u0305 . c\u0305) b\u0305 \u2013 (b\u0305 . c\u0305) a\u0305 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-27-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-747\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-748\"><span class=\"mfrac\" id=\"MathJax-Span-749\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-750\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-751\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-27\">\\frac{1}{3}<\/script>|b\u0305||c\u0305|a\u0305<br> \u2235 a,b, c are non collinear vectors<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7239\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5c-4.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 4\" width=\"322\" height=\"343\" sizes=\"auto, (max-width: 322px) 100vw, 322px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 4\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c)\"><\/p>\n\n<h3>Question 4.<br> Find the volume of the tatrahedron whose vertices are (1, 2, 1), (3, 2, 5), (2. \u2013 1, 0) and (- 1, 0, 1). (Mar. 2015-T.S) [May 2007]<\/h3>\n\n<p>Answer:<br> Let O be the origin with A, B, C, D as vertices of tetrahedron.<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7238\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5c-5.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 5\" width=\"292\" height=\"405\" sizes=\"auto, (max-width: 292px) 100vw, 292px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 5\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c)\"><\/p>\n\n\n\n<h3>Question 5.<br> Show that (a\u0305 + b\u0305) . (b\u0305 + c\u0305) \u00d7 (c\u0305 + a\u0305) = 2 [a\u0305 b\u0305 c\u0305]<\/h3>\n\n<p>Answer:<br> (a\u0305 + b\u0305) .(b\u0305 + c\u0305) \u00d7 (c\u0305 + a\u0305)<br> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-28-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-752\" style=\"width: 5.128em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.429em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1004.29em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-753\"><span class=\"mrow\" id=\"MathJax-Span-754\"><span class=\"mo\" id=\"MathJax-Span-755\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-756\"><span style=\"display: inline-block; position: relative; width: 3.497em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-757\"><span class=\"mrow\" id=\"MathJax-Span-758\"><span class=\"mn\" id=\"MathJax-Span-759\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-766\"><span class=\"mrow\" id=\"MathJax-Span-767\"><span class=\"mn\" id=\"MathJax-Span-768\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-775\"><span class=\"mrow\" id=\"MathJax-Span-776\"><span class=\"mn\" id=\"MathJax-Span-777\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-760\"><span class=\"mrow\" id=\"MathJax-Span-761\"><span class=\"mn\" id=\"MathJax-Span-762\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-769\"><span class=\"mrow\" id=\"MathJax-Span-770\"><span class=\"mn\" id=\"MathJax-Span-771\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-778\"><span class=\"mrow\" id=\"MathJax-Span-779\"><span class=\"mn\" id=\"MathJax-Span-780\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 2.985em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-763\"><span class=\"mrow\" id=\"MathJax-Span-764\"><span class=\"mn\" id=\"MathJax-Span-765\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-772\"><span class=\"mrow\" id=\"MathJax-Span-773\"><span class=\"mn\" id=\"MathJax-Span-774\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-781\"><span class=\"mrow\" id=\"MathJax-Span-782\"><span class=\"mn\" id=\"MathJax-Span-783\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-784\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-28\">\\left|\\begin{array}{lll}  1 & 1 & 0 \\\\  0 & 1 & 1 \\\\  1 & 0 & 1  \\end{array}\\right|<\/script>[a\u0305 b\u0305 c\u0305]<br> = [1(1) \u2013 1(-1)][a\u0305 b\u0305 c\u0305] = 2[a\u0305 b\u0305 c\u0305]<\/p>\n\n<h3>Question 6.<br> Show that the equation of the plane passing through the points with position vectors 3i\u0305 \u2013 5j\u0305 \u2013 k\u0305, -i\u0305 + 5j\u0305 + k\u0305 and parallel to the vector 3i\u0305 \u2013 j\u0305 + 7k\u0305 is 3x + 2y \u2013 z = 0.<\/h3>\n\n<p>Answer:<br> Let <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-29-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-785\" style=\"width: 1.82em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.54em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.074em, 1001.54em, 2.425em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-786\"><span class=\"munderover\" id=\"MathJax-Span-787\"><span style=\"display: inline-block; position: relative; width: 1.54em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.49em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-788\"><span class=\"mrow\" id=\"MathJax-Span-789\"><span class=\"mi\" id=\"MathJax-Span-790\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-791\" style=\"font-family: MathJax_Main;\">A<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.49em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-792\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.214em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.515em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.701em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.888em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.074em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.051em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-29\">\\overline{\\mathrm{OA}}<\/script> = (3i\u0305 \u2013 5 j\u0305 \u2013 k\u0305), <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-30-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-793\" style=\"width: 1.726em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1001.49em, 2.425em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-794\"><span class=\"munderover\" id=\"MathJax-Span-795\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.45em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-796\"><span class=\"mrow\" id=\"MathJax-Span-797\"><span class=\"mi\" id=\"MathJax-Span-798\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-799\" style=\"font-family: MathJax_Main;\">B<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-800\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.375em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.795em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.051em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-30\">\\overline{\\mathrm{OB}}<\/script> = \u2013 i\u0305 + 5j\u0305 + 7k\u0305<br> The given plane passes through the points A, B and parallel to the vector<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-31-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-801\" style=\"width: 1.726em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1001.49em, 2.425em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-802\"><span class=\"munderover\" id=\"MathJax-Span-803\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.45em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-804\"><span class=\"mrow\" id=\"MathJax-Span-805\"><span class=\"mi\" id=\"MathJax-Span-806\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-807\" style=\"font-family: MathJax_Main;\">C<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.49em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-808\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.282em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.468em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.655em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.841em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.027em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.051em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-31\">\\overline{\\mathrm{OC}}<\/script> = 3i\u0305 \u2013 j\u0305 + 7k\u0305,<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-32-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-809\" style=\"width: 8.25em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 7.085em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1007.09em, 2.565em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-810\"><span class=\"munderover\" id=\"MathJax-Span-811\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.4em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-812\"><span class=\"mrow\" id=\"MathJax-Span-813\"><span class=\"mi\" id=\"MathJax-Span-814\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-815\" style=\"font-family: MathJax_Main;\">B<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-816\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.121em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.748em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-817\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-818\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.45em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-819\"><span class=\"mrow\" id=\"MathJax-Span-820\"><span class=\"mi\" id=\"MathJax-Span-821\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-822\" style=\"font-family: MathJax_Main;\">B<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-823\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.167em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.142em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.375em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.562em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.795em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.981em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-824\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"munderover\" id=\"MathJax-Span-825\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 1.54em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.49em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-826\"><span class=\"mrow\" id=\"MathJax-Span-827\"><span class=\"mi\" id=\"MathJax-Span-828\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-829\" style=\"font-family: MathJax_Main;\">A<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.49em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-830\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.493em; height: 0px;\"><span style=\"position: absolute; top: -4.005em; left: -0.044em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.214em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.096em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.329em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.515em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.701em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 0.888em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.005em; left: 1.074em;\"><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.159em; border-left: 0px solid; width: 0px; height: 1.408em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-32\">\\overline{\\mathrm{AB}}=\\overline{\\mathrm{OB}}-\\overline{\\mathrm{OA}}<\/script> = -4i\u0305 + 10j\u0305 + 8k\u0305<br> \u2234 Equation of the plane is<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7237\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5c-6.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 6\" width=\"320\" height=\"230\" sizes=\"auto, (max-width: 320px) 100vw, 320px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 6\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c)\"><br> = x (70 + 8) \u2013 y (- 28 \u2013 24) + z (4 \u2013 30)<br> = 78x + 52y \u2013 26z<br> = 26 (3x + 2y \u2013 z)<\/p>\n\n<p><span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-33-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-831\" style=\"width: 14.681em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 12.63em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.192em, 1012.49em, 6.247em, -999.998em); top: -4.471em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-832\"><span class=\"mrow\" id=\"MathJax-Span-833\"><span class=\"mo\" id=\"MathJax-Span-834\" style=\"\"><span style=\"font-family: MathJax_Main;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-835\"><span style=\"display: inline-block; position: relative; width: 3.59em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.51em, 4.243em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.264em, 1000.51em, 4.15em, -999.998em); top: -3.912em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-836\"><span class=\"mrow\" id=\"MathJax-Span-837\"><span class=\"texatom\" id=\"MathJax-Span-838\"><span class=\"mrow\" id=\"MathJax-Span-839\"><span class=\"munderover\" id=\"MathJax-Span-840\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-841\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1000.42em, 3.59em, -999.998em); top: -4.005em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-842\" style=\"font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.078em, 1000.47em, 4.243em, -999.998em); top: -4.005em; left: 1.54em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1000.47em, 4.15em, -999.998em); top: -3.912em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-843\"><span class=\"mrow\" id=\"MathJax-Span-844\"><span class=\"texatom\" id=\"MathJax-Span-845\"><span class=\"mrow\" id=\"MathJax-Span-846\"><span class=\"munderover\" id=\"MathJax-Span-847\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0.049em;\"><span class=\"mi\" id=\"MathJax-Span-848\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1000.42em, 3.59em, -999.998em); top: -4.285em; left: 0em;\"><span class=\"mo\" id=\"MathJax-Span-849\" style=\"font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.47em, 4.243em, -999.998em); top: -4.005em; left: 3.031em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.264em, 1000.47em, 4.15em, -999.998em); top: -3.912em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-850\"><span class=\"mrow\" id=\"MathJax-Span-851\"><span class=\"texatom\" id=\"MathJax-Span-852\"><span class=\"mrow\" id=\"MathJax-Span-853\"><span class=\"munderover\" id=\"MathJax-Span-854\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0.049em;\"><span class=\"mi\" id=\"MathJax-Span-855\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1000.42em, 3.59em, -999.998em); top: -4.005em; left: 0.049em;\"><span class=\"mo\" id=\"MathJax-Span-856\" style=\"font-family: MathJax_Main;\">\u00af<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-857\" style=\"\"><span style=\"font-family: MathJax_Main;\">]<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-858\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mrow\" id=\"MathJax-Span-859\" style=\"padding-left: 0.282em;\"><span class=\"mo\" id=\"MathJax-Span-860\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-861\"><span style=\"display: inline-block; position: relative; width: 5.827em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1001.26em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-862\"><span class=\"mrow\" id=\"MathJax-Span-863\"><span class=\"mn\" id=\"MathJax-Span-864\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1001.26em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-873\"><span class=\"mrow\" id=\"MathJax-Span-874\"><span class=\"mo\" id=\"MathJax-Span-875\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-876\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-883\"><span class=\"mrow\" id=\"MathJax-Span-884\"><span class=\"mn\" id=\"MathJax-Span-885\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-865\"><span class=\"mrow\" id=\"MathJax-Span-866\"><span class=\"mo\" id=\"MathJax-Span-867\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-868\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.98em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-877\"><span class=\"mrow\" id=\"MathJax-Span-878\"><span class=\"mn\" id=\"MathJax-Span-879\" style=\"font-family: MathJax_Main;\">10<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-886\"><span class=\"mrow\" id=\"MathJax-Span-887\"><span class=\"mo\" id=\"MathJax-Span-888\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-889\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.107em, -999.998em); top: -4.471em; left: 4.569em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-869\"><span class=\"mrow\" id=\"MathJax-Span-870\"><span class=\"mo\" id=\"MathJax-Span-871\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-872\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-880\"><span class=\"mrow\" id=\"MathJax-Span-881\"><span class=\"mn\" id=\"MathJax-Span-882\" style=\"font-family: MathJax_Main;\">8<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-890\"><span class=\"mrow\" id=\"MathJax-Span-891\"><span class=\"mn\" id=\"MathJax-Span-892\" style=\"font-family: MathJax_Main;\">7<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-893\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-33\">\\left[\\begin{array}{lll}  \\bar{a} & \\bar{b} & \\bar{c}  \\end{array}\\right]=\\left|\\begin{array}{rrr}  3 & -5 & -1 \\\\  -4 & 10 & 8 \\\\  3 & -1 & 7  \\end{array}\\right|<\/script><br> = 3 (70 + 8) + 5 (- 28 \u2013 24) \u2013 1 (4 \u2013 30)<br> = 234 \u2013 260 + 26 = 0<br> Equation of the required plane is<br> 26 (3x + 2y \u2013 z) = 0<br> \u21d2 3x + 2y \u2013 z = 0<\/p>\n\n\n\n<h3>Question 7.<br> Prove that a\u0305 \u00d7 [a\u0305 \u00d7 (a\u0305 \u00d7 -(a\u0305 . a\u0305) (b\u0305 \u00d7 a\u0305)<\/h3>\n\n<p>Answer:<br> L.H.S = a \u00d7 [a\u0305 \u00d7 (a\u0305 \u00d7 b\u0305)]<br> = a\u0305 \u00d7 [(a\u0305 . b\u0305)a \u2013 (a\u0305. a\u0305)b\u0305]<br> = (a\u0305. b\u0305) (a\u0305 \u00d7 a\u0305) \u2013 (a\u0305. a\u0305) (a\u0305 \u00d7 b\u0305)<br> = (a\u0305. b\u0305) (0) \u2013 (a\u0305. a\u0305) (a\u0305 \u00d7 b\u0305)<br> = (a\u0305. a\u0305) (b\u0305 \u00d7 a\u0305)<br> = R.H.S.<\/p>\n\n<h3>Question 8.<br> If a\u0305, b\u0305, c\u0305 and d\u0305 are coplanar vectors, then show that (a\u0305 \u00d7 b\u0305) \u00d7 (c\u0305 \u00d7 d\u0305) = 0.<\/h3>\n\n<p>Answer:<br> Given a\u0305, b\u0305, c\u0305, d\u0305 are coplanar vectors<br> \u21d2 a\u0305 \u00d7 b\u0305 is perpendicular to the plane S.<br> In the similar way c\u0305 \u00d7 d\u0305 is perpendicular to the plane S.<br> a\u0305 \u00d7 b\u0305 and c\u0305 \u00d7 d\u0305<br> are parallel vectors.<br> \u21d2 (a\u0305 \u00d7 b\u0305) \u00d7 (c\u0305 \u00d7 d\u0305) = 0 (or)<br> (a\u0305 \u00d7 b\u0305) \u00d7 (c\u0305 \u00d7 d\u0305) = [a\u0305 c\u0305 d\u0305]b\u0305 \u2013 [b\u0305 c\u0305 d\u0305]a\u0305<br> = 0b\u0305 \u2013 0a\u0305 = 0 (\u2235 a\u0305, b\u0305, c\u0305, d\u0305 are coplanar)<\/p>\n\n<h3>Question 9.<br> Show that [(a\u0305 \u00d7 b\u0305) \u00d7 (a\u0305 \u00d7 c\u0305)] d\u0305 = (a\u0305 . d\u0305) [a\u0305 b\u0305 c\u0305]<\/h3>\n\n<p>Answer:<br> We have (a\u0305 \u00d7 b\u0305) \u00d7 (c\u0305 \u00d7 d\u0305)<br> = [a\u0305 c\u0305 d\u0305]b\u0305 \u2013 [b\u0305 c\u0305 d\u0305]a\u0305<br> (a\u0305 \u00d7 b\u0305) \u00d7 (a\u0305 \u00d7 c\u0305) = [a\u0305 a\u0305 c\u0305] b\u0305 \u2013 [b\u0305 a\u0305 c\u0305]a\u0305<br> = 0(b\u0305) \u2013 [b\u0305 a\u0305 c\u0305] a\u0305 = (a\u0305 b\u0305 c\u0305)a\u0305<br> [(a\u0305 \u00d7 b\u0305) \u00d7 (a\u0305 \u00d7 c\u0305)] d\u0305 = [a\u0305 b\u0305 c\u0305](a\u0305.d\u0305)<\/p>\n\n<h3>Question 10.<br> Show that a\u0305.[(b\u0305 + c\u0305) \u00d7 [a\u0305 + b\u0305 + c\u0305]] = 0<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7236\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5c-7.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 7\" width=\"279\" height=\"461\" sizes=\"auto, (max-width: 279px) 100vw, 279px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 7\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c)\"><\/p>\n\n <h3>Question 11.<br> Find \u03bb in order that the four points A (3, 2, 1), B (4, \u03bb, 5), C (4, 2, \u2013 2) and D (6, 5, \u2013 1) are coplanar.<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7235\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5c-8.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 8\" width=\"316\" height=\"322\" sizes=\"auto, (max-width: 316px) 100vw, 316px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 8\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c)\"><br> \u21d2 1(9) \u2013 (\u03bb \u2013 2) (- 2 + 9) + 4 (3) = 0<br> \u21d2 21 \u2013 (\u03bb \u2013 2)(7) = 0<br> \u21d2 \u03bb \u2013 2 = 3<br> \u21d2 \u03bb = 5<\/p>\n\n<h3>Question 12.<br> Find the vector equation of the plane passing through the intersection of planes.<br> r\u0305 -(2i\u0305 + 2j\u0305 \u2013 3k\u0305) = 7, r\u0305 = (2i\u0305 + 5j\u0305 + 3k\u0305) = 9 and through the point (2, 1, 3).<\/h3>\n\n<p>Answer:<br> The planes are of the form<br> r\u0305.n\u0305<sub>1<\/sub> = d<sub>1<\/sub> and r\u0305.n\u0305<sub>2<\/sub> = d<sub>2<\/sub><br> The vector equation of the plane passing through the intersection of above plane is of the form<br> r\u0305 . (n\u0305<sub>1<\/sub> + \u03bbn\u0305<sub>2<\/sub>) = d<sub>1<\/sub> + \u03bbd<sub>2<\/sub><br> \u2234 r\u0305 . [(2i\u0305 + 2j\u0305 \u2013 3k\u0305) + \u03bb(2i\u0305 + 5j\u0305 + 3k\u0305)]<br> = 7 + 9\u03bb<\/p>\n\n<p>Denote r\u0305 = xi\u0305 + yj\u0305 + zk\u0305 \u2026\u2026\u2026..(1)<br> then (2x + 2y \u2013 3z) + \u03bb (2x + 5y + 3z)] = 7 + 9\u03bb<br> \u21d2 (2x + 2y \u2013 3z \u2013 7) + \u03bb (2x + 5y + 3z \u2013 9) = 0 \u2026..(2)<br> Since this plane passes through the point (2, 1, 3)<br> We have<br> (4 + 2 \u2013 9 \u2013 7) + \u03bb (4 + 5 + 9 \u2013 9) = 0<br> \u21d2 \u2013 10 + 9\u03bb = 0<br> \u21d2 \u03bb = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-34-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-894\" style=\"width: 1.307em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.121em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.12em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-895\"><span class=\"mfrac\" id=\"MathJax-Span-896\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.7em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-897\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">10<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-898\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">9<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.84em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.841em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-34\">\\frac{10}{9}<\/script><br> From (2), (2 + 2\u03bb) x + (2 + 5\u03bb) y + (3\u03bb \u2013 3) z \u2013 (7 + 9\u03bb) = 0<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7234\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5c-9.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 9\" width=\"327\" height=\"203\" sizes=\"auto, (max-width: 327px) 100vw, 327px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 9\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c)\"><\/p>\n\n<h3>Question 13.<br> Find the equation of the plane passing through (a\u0305, b\u0305, c\u0305) and parallel to the plane r\u0305. (i\u0305 + j\u0305 + k\u0305) = 2.<\/h3>\n\n<p>Answer:<br> Given equation of the plane is<br> r\u0305 . (i\u0305 + j\u0305 + k\u0305) = 2<br> Suppose r\u0305 = xi\u0305 + yj\u0305 + zk\u0305 then<br> (xi\u0305 + yj\u0305 + zk\u0305) . (i\u0305 + j\u0305 + k\u0305) = 2<br> \u21d2 x + y + z = 2<br> Equation of parallel plane is x + y + z = k<br> Since this passes through (a, b, c) we have a + b + c = k<br> Equation of the required plane is x + y + z = a + b + c<\/p>\n\n<h3>Question 14.<br> Find the shortest distance between the lines r\u0305 = 6i\u0305 + 2j\u0305 + 2k\u0305 + \u03bb(i\u0305 \u2013 2j\u0305 + 2k\u0305) and r\u0305 = -4i\u0305 \u2013 k\u0305 + \u03bc(3i\u0305 \u2013 2j\u0305 \u2013 2k\u0305).<\/h3>\n\n<p>Answer:<br> Given lines are<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7233\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5c-10.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 10\" width=\"327\" height=\"800\" sizes=\"auto, (max-width: 327px) 100vw, 327px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 10\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c)\"><br> Shortest distance between the lines = 9 units.<\/p>\n\n\n\n<h3>Question 15.<br> Find the equation of the plane passing through the line of intersection of the planes<br> r\u0305.(i\u0305 + j\u0305 + k\u0305) = l and r\u0305.(2i\u0305 + 3j\u0305 \u2013 k\u0305) + 4 = 0 and parallel to X \u2013 axis.<\/h3>\n\n<p>Answer:<br> Cartesian form of the given plane is x + y + z = 1 and 2x + 3y \u2013 z + 4 = 0<br> Equation of required plane will be of the form<br> (x + y + z \u2013 1) + \u03bb (2x + 3y \u2013 z + 4) = 0 \u2026\u2026\u2026\u2026(i)<br> \u21d2 (1 + 2\u03bb)x + (1 + 3\u03bb)y + (1 \u2013 \u03bb)z \u2013 (1 \u2013 4\u03bb) = 0<br> Since this is parallel to X-axis coefficient of x = 0<br> \u21d2 1 + 2\u03bb = 0 \u21d2 \u03bb = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-35-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-899\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-900\"><span class=\"mfrac\" id=\"MathJax-Span-901\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-902\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-903\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-35\">\\frac{1}{2}<\/script><br> Required plane equation from (1) is<br> (x + y + z \u2013 1) \u2013<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-36-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-904\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-905\"><span class=\"mfrac\" id=\"MathJax-Span-906\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-907\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-908\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-36\">\\frac{1}{2}<\/script>(2x + 3y \u2013 z + 4) = 0<br> \u21d2 2x + 2y + 2z \u2013 2 \u2013 2x \u2013 3y + z \u2013 4 = 0<br> \u21d2 y \u2013 3z + 6 = 0<\/p>\n\n <h3>Question 16.<br> Prove that the four points 4i\u0305 + 5j\u0305 + k\u0305, -(j\u0305 + k\u0305), 3i\u0305 + 9j\u0305 + 4k\u0305 and -4i\u0305 + 4j\u0305 + 4k\u0305 are coplanar.<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7232\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5c-11.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 11\" width=\"266\" height=\"353\" sizes=\"auto, (max-width: 266px) 100vw, 266px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 11\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c)\"><br> = \u2013 4 (12 + 3) + 6 (- 3 + 24) \u2013 2(1 + 32)<br> = -60 + 126 \u2013 66 = 126 \u2013 126 = 0<br> Given points are coplanar.<\/p>\n\n<h3>Question 17.<br> If a\u0305, b\u0305, c\u0305 are non coplanar, then show that the vectors a\u0305 \u2013 b\u0305, b\u0305 + c\u0305, c\u0305 + a\u0305 are coplanar.<\/h3>\n\n<p>Answer:<br> Given that a\u0305, b\u0305, c\u0305 are non coplanar<br> we have [a\u0305 b\u0305 c\u0305] ^O<br> \u2234 [a\u0305 \u2013 b\u0305 b\u0305 + c\u0305 c\u0305 + a\u0305]<br> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-37-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-909\" style=\"width: 6.06em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.221em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.08em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-910\"><span class=\"mrow\" id=\"MathJax-Span-911\"><span class=\"mo\" id=\"MathJax-Span-912\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-913\"><span style=\"display: inline-block; position: relative; width: 4.289em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-914\"><span class=\"mrow\" id=\"MathJax-Span-915\"><span class=\"mn\" id=\"MathJax-Span-916\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-924\"><span class=\"mrow\" id=\"MathJax-Span-925\"><span class=\"mn\" id=\"MathJax-Span-926\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-933\"><span class=\"mrow\" id=\"MathJax-Span-934\"><span class=\"mn\" id=\"MathJax-Span-935\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.107em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-917\"><span class=\"mrow\" id=\"MathJax-Span-918\"><span class=\"mo\" id=\"MathJax-Span-919\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-920\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-927\"><span class=\"mrow\" id=\"MathJax-Span-928\"><span class=\"mn\" id=\"MathJax-Span-929\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-936\"><span class=\"mrow\" id=\"MathJax-Span-937\"><span class=\"mn\" id=\"MathJax-Span-938\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-921\"><span class=\"mrow\" id=\"MathJax-Span-922\"><span class=\"mn\" id=\"MathJax-Span-923\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-930\"><span class=\"mrow\" id=\"MathJax-Span-931\"><span class=\"mn\" id=\"MathJax-Span-932\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-939\"><span class=\"mrow\" id=\"MathJax-Span-940\"><span class=\"mn\" id=\"MathJax-Span-941\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-942\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-37\">\\left|\\begin{array}{ccc}  1 & -1 & 0 \\\\  0 & 1 & 1 \\\\  1 & 0 & 1  \\end{array}\\right|<\/script>[a\u0305 b\u0305 c\u0305]<br> = [1 + 1 (-1)][a\u0305 b\u0305 c\u0305]<br> = 0 [a\u0305 b\u0305 c\u0305] = 0<br> \u2234 Vectors a\u0305 \u2013 b\u0305, b\u0305 + c\u0305, c\u0305 + a\u0305 are coplanar.<\/p>\n\n<h3>Question 18.<br> If a\u0305, b\u0305, c\u0305 are the position vectors of the points A, B and C respectively, then prove that the vector a\u0305 \u00d7 b\u0305 + b\u0305 \u00d7 c\u0305 + c\u0305 \u00d7 a\u0305 is perpendicular to the plane of \u0394ABC.<\/h3>\n\n<p>Answer:<br> Let O be the origin and<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7231\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5c-12.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 12\" width=\"329\" height=\"239\" sizes=\"auto, (max-width: 329px) 100vw, 329px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c) 12\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(c)\"><br> = (b\u0305 \u00d7 c\u0305) \u2013 (b\u0305 \u00d7 a\u0305) \u2013 (a\u0305 \u00d7 c\u0305) + (a\u0305 \u00d7 a\u0305)<br> = (b\u0305 \u00d7 c\u0305) + (a\u0305 \u00d7 b\u0305) + (c\u0305 \u00d7 a\u0305) (\u2235 (a\u0305 \u00d7 a\u0305) = 0)<br> = (a\u0305 \u00d7 b\u0305) + (b\u0305 \u00d7 c\u0305) + (c\u0305 \u00d7 a\u0305)<br> Hence (a\u0305 \u00d7 b\u0305) + (b\u0305 \u00d7 c\u0305) + (c\u0305 \u00d7 a\u0305) is perpendicular to the plane of \u0394ABC.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>I. Question 1. Compute [i\u0305 \u2013 j\u0305 j\u0305 \u2013 k\u0305 k\u0305 \u2013 i\u0305] Answer: [i\u0305 \u2013 j\u0305 j\u0305 \u2013 k\u0305 k\u0305 \u2013 i\u0305] = \u2223\u2223\u2223\u222310\u22121\u22121100\u221211\u2223\u2223\u2223\u2223 = 1 (1) + 1 (- 1) = 1 \u2013 1 = 0 Question 2. If a\u0305 = i\u0305 \u2013 2j\u0305 \u2013 3k\u0305, b\u0305 = 2i\u0305 + j\u0305 \u2013 k\u0305, [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":3056,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1544,15,38],"tags":[],"class_list":{"0":"post-3018","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-study-material-tg-inter","8":"category-telangana","9":"category-tg-inter"},"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\r\n<title>TS Inter 1st Year Maths 1A Products of Vectors Solutions Exercise 5(C)<\/title>\r\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\r\n<link rel=\"canonical\" href=\"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5c\/\" \/>\r\n<meta property=\"og:locale\" content=\"en_US\" \/>\r\n<meta property=\"og:type\" content=\"article\" \/>\r\n<meta property=\"og:title\" content=\"TS Inter 1st Year Maths 1A Products of Vectors Solutions Exercise 5(C)\" \/>\r\n<meta property=\"og:description\" content=\"I. Question 1. Compute [i\u0305 \u2013 j\u0305 j\u0305 \u2013 k\u0305 k\u0305 \u2013 i\u0305] Answer: [i\u0305 \u2013 j\u0305 j\u0305 \u2013 k\u0305 k\u0305 \u2013 i\u0305] = \u2223\u2223\u2223\u222310\u22121\u22121100\u221211\u2223\u2223\u2223\u2223 = 1 (1) + 1 (- 1) = 1 \u2013 1 = 0 Question 2. If a\u0305 = i\u0305 \u2013 2j\u0305 \u2013 3k\u0305, b\u0305 = 2i\u0305 + j\u0305 \u2013 k\u0305, [&hellip;]\" \/>\r\n<meta property=\"og:url\" content=\"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5c\/\" \/>\r\n<meta property=\"og:site_name\" content=\"Manabadi Boards\" \/>\r\n<meta property=\"article:published_time\" content=\"2026-01-21T06:43:59+00:00\" \/>\r\n<meta property=\"article:modified_time\" content=\"2026-01-21T11:20:59+00:00\" \/>\r\n<meta property=\"og:image\" content=\"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114618\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5c.jpg\" \/>\r\n\t<meta property=\"og:image:width\" content=\"900\" \/>\r\n\t<meta property=\"og:image:height\" content=\"600\" \/>\r\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\r\n<meta name=\"author\" content=\"Junaid\" \/>\r\n<meta name=\"twitter:card\" 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Question 1. Compute [i\u0305 \u2013 j\u0305 j\u0305 \u2013 k\u0305 k\u0305 \u2013 i\u0305] Answer: [i\u0305 \u2013 j\u0305 j\u0305 \u2013 k\u0305 k\u0305 \u2013 i\u0305] = \u2223\u2223\u2223\u222310\u22121\u22121100\u221211\u2223\u2223\u2223\u2223 = 1 (1) + 1 (- 1) = 1 \u2013 1 = 0 Question 2. 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