{"id":2978,"date":"2026-01-20T12:23:51","date_gmt":"2026-01-20T06:53:51","guid":{"rendered":"https:\/\/www.manabadi.co.in\/boards\/?p=2978"},"modified":"2026-01-20T18:10:57","modified_gmt":"2026-01-20T12:40:57","slug":"ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5b-2","status":"publish","type":"post","link":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5b-2\/","title":{"rendered":"TS Inter 1st Year Maths 1A Products of Vectors Solutions Exercise 5(b)"},"content":{"rendered":"\n<h3>I.<br> Question 1.<br> If |p\u0305| = 2, |q\u0305| = 3 and (p, q) = &#960;\/6 , then find |p\u0305 \u00d7 q\u0305|<sup>2<\/sup>.<\/h3>\n\n<p>Answer:<br> p\u0305 \u00d7 q\u0305 = |p\u0305| |q\u0305| sin\u03b8n\u0302<br> Given p\u0305 = 2, q\u0305 = 3 and (p\u0305. q\u0305) = &#960;\/6 <br> |p\u0305 \u00d7 q\u0305| = (2) (3)sin &#960;\/6 =3<br> \u2234 |p\u0305 \u00d7 q\u0305|<sup>2<\/sup> = 9<\/p>\n\n<h3>Question 2.<br> If a\u0305 = 2i\u0305 \u2013 j\u0305 + k\u0305 and b\u0305 = i\u0305 \u2013 3j\u0305 \u2013 5k\u0305, then find |a\u0305 \u00d7 b\u0305|. (March 2013)<\/h3>\n\n<p>Answer:<br> a\u0305 = 2 i\u0305 \u2013 j\u0305 + k\u0305 and b\u0305 = i\u0305 \u2013 3 j\u0305 \u2013 5k\u0305<br> <img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-7098\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5b-1.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(b) 1\" width=\"272\" height=\"204\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(b) 1\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(b)\"><\/p>\n\n<h3>Question 3.<br> If a\u0305 = 2i\u0305 \u2013 3j\u0305 + k\u0305 and b\u0305 = i\u0305 + 4j\u0305 \u2013 2k\u0305, then find (a\u0305 + b\u0305) \u00d7 (a\u0305 \u2013 b\u0305).<\/h3>\n\n<p>Answer:<br> Given a\u0305 = 2i\u0305 \u2013 3j\u0305 + k\u0305 and b\u0305 = i\u0305 + 4j\u0305 \u2013 2k\u0305<br> Then a\u0305 + b\u0305 = 3 i\u0305 + j\u0305 \u2013 k\u0305 and a\u0305 \u2013 b\u0305 = i\u0305 \u2013 7j\u0305 + 3k\u0305<br> (a + b) \u00d7 (a \u2013 b) = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-4-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-16\" 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, 6.479em, -999.998em); top: -4.564em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-17\"><span class=\"mrow\" id=\"MathJax-Span-18\"><span class=\"mo\" id=\"MathJax-Span-19\" style=\"vertical-align: 2.286em;\"><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: -0.231em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.722em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -0.976em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-20\"><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(2.425em, 1000.47em, 6.526em, -999.998em); top: -4.797em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.891em, 1000.33em, 4.15em, -999.998em); top: -5.263em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-21\"><span class=\"mrow\" id=\"MathJax-Span-22\"><span class=\"munderover\" id=\"MathJax-Span-23\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.24em, 4.15em, -999.998em); top: -4.005em; left: 0.049em;\"><span class=\"texatom\" id=\"MathJax-Span-24\"><span class=\"mrow\" id=\"MathJax-Span-25\"><span class=\"mi\" id=\"MathJax-Span-26\" style=\"font-family: MathJax_Main;\">i<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.33em, 3.777em, -999.998em); top: -4.471em; left: 0em;\"><span class=\"mo\" id=\"MathJax-Span-27\" style=\"\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/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 style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.819em; right: 0em;\"><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;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.421em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-52\"><span class=\"mrow\" id=\"MathJax-Span-53\"><span class=\"mn\" id=\"MathJax-Span-54\" 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: 4.802em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.519em, 1001.26em, 6.666em, -999.998em); top: -4.89em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.891em, 1000.33em, 4.336em, -999.998em); top: -5.263em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-28\"><span class=\"mrow\" id=\"MathJax-Span-29\"><span class=\"munderover\" id=\"MathJax-Span-30\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.24em, 4.336em, -999.998em); top: -4.005em; left: 0.049em;\"><span class=\"texatom\" id=\"MathJax-Span-31\"><span class=\"mrow\" id=\"MathJax-Span-32\"><span class=\"mi\" id=\"MathJax-Span-33\" style=\"font-family: MathJax_Main;\">j<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.33em, 3.777em, -999.998em); top: -4.471em; left: 0em;\"><span class=\"mo\" id=\"MathJax-Span-34\" style=\"\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/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 style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.819em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-45\"><span class=\"mrow\" id=\"MathJax-Span-46\"><span class=\"mn\" id=\"MathJax-Span-47\" 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, 1001.26em, 4.243em, -999.998em); top: -2.421em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-55\"><span class=\"mrow\" id=\"MathJax-Span-56\"><span class=\"mo\" id=\"MathJax-Span-57\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-58\" 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: 4.895em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.21em, 6.573em, -999.998em); top: -4.844em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.845em, 1000.51em, 4.15em, -999.998em); top: -5.263em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-35\"><span class=\"mrow\" id=\"MathJax-Span-36\"><span class=\"munderover\" id=\"MathJax-Span-37\"><span style=\"display: inline-block; position: relative; width: 0.515em; 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-38\"><span class=\"mrow\" id=\"MathJax-Span-39\"><span class=\"mi\" id=\"MathJax-Span-40\" style=\"font-family: MathJax_Main;\">k<\/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-41\" 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: 0.189em;\"><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 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 style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.819em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-48\"><span class=\"mrow\" id=\"MathJax-Span-49\"><span class=\"mo\" id=\"MathJax-Span-50\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-51\" 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: -2.421em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-59\"><span class=\"mrow\" id=\"MathJax-Span-60\"><span class=\"mn\" id=\"MathJax-Span-61\" 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: 4.849em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-62\" style=\"vertical-align: 2.286em;\"><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: -0.231em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.722em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -0.976em; 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: 4.569em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -2.105em; border-left: 0px solid; width: 0px; height: 4.814em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-4\">\\left|\\begin{array}{rrr}  \\overline{\\mathrm{i}} & \\overline{\\mathrm{j}} & \\overline{\\mathrm{k}} \\\\  3 & 1 & -1 \\\\  1 & -7 & 3  \\end{array}\\right|<\/script><br> = i\u0305(3 \u2013 7) \u2013 j\u0305(9 + l) + k\u0305 (- 21 \u2013 1)<br> = -4i\u0305 \u2013 10j\u0305 \u2013 22k\u0305<br> = -2 (2i\u0305 + 5j\u0305 + 11k\u0305)<\/p>\n\n<h3>Question 4.<br> If 4i\u0305 + <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-5-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-63\" 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(1.121em, 1001.12em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-64\"><span class=\"mfrac\" id=\"MathJax-Span-65\"><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, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-66\"><span class=\"mn\" id=\"MathJax-Span-67\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-68\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">p<\/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-69\" 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, 1000.84em, 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: 1.732em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-5\">\\frac{2 p}{3}<\/script> j\u0305 + pk\u0305 is parallel to the vector 3 i\u0305 + 2j\u0305 + 3k\u0305, find p.<\/h3>\n\n<p>Answer:<br> Given 4i\u0305 + <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-6-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-70\" 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(1.121em, 1001.12em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-71\"><span class=\"mfrac\" id=\"MathJax-Span-72\"><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, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-73\"><span class=\"mn\" id=\"MathJax-Span-74\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-75\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">p<\/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-76\" 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, 1000.84em, 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: 1.732em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-6\">\\frac{2 p}{3}<\/script> j\u0305 + pk\u0305 is parallel to<br> i\u0305 + 2j\u0305 + 3k\u0305<br> \u2234 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-7-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-77\" style=\"width: 6.34em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.454em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(0.748em, 1005.45em, 2.845em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-78\"><span class=\"mfrac\" id=\"MathJax-Span-79\"><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, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-80\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-81\" 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(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 class=\"mo\" id=\"MathJax-Span-82\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-83\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.124em, 1000.84em, 4.476em, -999.998em); top: -4.704em; left: 50%; margin-left: -0.417em;\"><span class=\"mfrac\" id=\"MathJax-Span-84\"><span style=\"display: inline-block; position: relative; width: 0.608em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.51em, 4.243em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.231em;\"><span class=\"mrow\" id=\"MathJax-Span-85\"><span class=\"mn\" id=\"MathJax-Span-86\" style=\"font-size: 50%; font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-87\" style=\"font-size: 50%; font-family: MathJax_Math-italic;\">p<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.544em, 1000.24em, 4.15em, -999.998em); top: -3.679em; left: 50%; margin-left: -0.137em;\"><span class=\"mn\" id=\"MathJax-Span-88\" style=\"font-size: 50%; 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, 1000.61em, 1.214em, -999.998em); top: -1.209em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.608em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/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-89\" 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, 1000.98em, 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: 0.981em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-90\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-91\" style=\"padding-left: 0.282em;\"><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(3.544em, 1000.38em, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-92\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">p<\/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-93\" 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: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 2.273em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-7\">\\frac{4}{1}=\\frac{\\frac{2 p}{3}}{2}=\\frac{p}{3}<\/script><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-94\" 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(1.121em, 1001.12em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-95\"><span class=\"mfrac\" id=\"MathJax-Span-96\"><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, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-97\"><span class=\"mn\" id=\"MathJax-Span-98\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-99\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">p<\/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-100\" 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, 1000.84em, 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: 1.732em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-8\">\\frac{2 p}{3}<\/script> = -4 \u21d2 p = 12<\/p>\n\n<h3>Question 5.<br> Compute<br> a\u0305 \u00d7 (b\u0305 + c\u0305) + b\u0305 \u00d7 (c\u0305 + a\u0305) + c\u0305 \u00d7 (a\u0305 + b\u0305)<\/h3>\n\n<p>Sol.<br> a\u0305 \u00d7 (b\u0305 + c\u0305) + b\u0305 \u00d7 (c\u0305 + a\u0305) + c\u0305 \u00d7 (a\u0305 + b\u0305)<br> = (a\u0305 \u00d7 b\u0305) + (a\u0305 \u00d7 c\u0305) + (b\u0305 \u00d7 c\u0305) + (b\u0305 \u00d7 a\u0305) + (c\u0305 \u00d7 a\u0305) + (c\u0305 \u00d7 b\u0305)<br> = (a\u0305 \u00d7 b\u0305) + (a\u0305 \u00d7 c\u0305) + (b\u0305 \u00d7 c\u0305) \u2013 (a\u0305 \u00d7 b\u0305) \u2013 (a\u0305 \u00d7 c\u0305) \u2013 (b\u0305 \u00d7 c\u0305)<br> = 0<\/p>\n\n<h3>Question 6.<br> If p\u0305 = xi\u0305 + yj\u0305 + zk\u0305, then find |p\u0305 \u00d7 k\u0305|<sup>2<\/sup>.<\/h3>\n\n<p>Answer:<br> p\u0305 \u00d7 k\u0305 = (xi\u0305 + yj\u0305 + zk\u0305) \u00d7 k\u0305<br> = x(i\u0305 \u00d7 k\u0305) + y(j\u0305 \u00d7 k\u0305) + z(k\u0305 \u00d7 k\u0305)<br> = -xj\u0305 + yi\u0305 + z(0)<br> = yi\u0305 \u2013 xj\u0305<br> |p\u0305 \u00d7 k\u0305|<sup>2<\/sup> = x<sup>2<\/sup> + y<sup>2<\/sup><\/p>\n\n<h3>Question 7.<br> Compute 2j\u0305 \u00d7 (3i\u0305 \u2013 4k\u0305) + (i\u0305 + 2j\u0305) \u00d7 k\u0305<\/h3>\n\n<p>Sol.<br> 2j\u0305 \u00d7 (3i\u0305 \u2013 4k\u0305) + (i\u0305 + 2j\u0305) \u00d7 k\u0305<br> = 6(j\u0305 \u00d7 i\u0305) \u2013 8(j\u0305 \u00d7 k\u0305) + (i\u0305 \u00d7 k\u0305) + 2(j\u0305 \u00d7 k\u0305)<br> = -6k\u0305 \u2013 8i\u0305 \u2013 j\u0305 + 2i\u0305<br> = -6i\u0305 \u2013 j\u0305 \u2013 6k\u0305<\/p>\n\n<h3>Question 8.<br> Find unit vector perpendicular to both i\u0305 + j\u0305 + k\u0305 and 2i\u0305 + j\u0305 + 3k\u0305.<\/h3>\n\n<p>Answer:<br> Given a\u0305 = i\u0305 + j\u0305 + k\u0305 and b\u0305 = 2i\u0305 + j\u0305 + 3k\u0305<br> then a\u0305 \u00d7 b\u0305 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-9-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-101\" style=\"width: 5.221em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.476em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1004.34em, 6.479em, -999.998em); top: -4.564em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-102\"><span class=\"mrow\" id=\"MathJax-Span-103\"><span class=\"mo\" id=\"MathJax-Span-104\" style=\"vertical-align: 2.286em;\"><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: -0.231em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.722em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -0.976em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-105\"><span style=\"display: inline-block; position: relative; width: 3.544em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.425em, 1000.47em, 6.526em, -999.998em); top: -4.797em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.891em, 1000.33em, 4.15em, -999.998em); top: -5.263em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-106\"><span class=\"mrow\" id=\"MathJax-Span-107\"><span class=\"munderover\" id=\"MathJax-Span-108\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.24em, 4.15em, -999.998em); top: -4.005em; left: 0.049em;\"><span class=\"texatom\" id=\"MathJax-Span-109\"><span class=\"mrow\" id=\"MathJax-Span-110\"><span class=\"mi\" id=\"MathJax-Span-111\" style=\"font-family: MathJax_Main;\">i<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.33em, 3.777em, -999.998em); top: -4.471em; left: 0em;\"><span class=\"mo\" id=\"MathJax-Span-112\" style=\"\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/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 style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.819em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-127\"><span class=\"mrow\" id=\"MathJax-Span-128\"><span class=\"mn\" id=\"MathJax-Span-129\" 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: -2.421em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-136\"><span class=\"mrow\" id=\"MathJax-Span-137\"><span class=\"mn\" id=\"MathJax-Span-138\" 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: 4.802em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1000.42em, 6.526em, -999.998em); top: -4.797em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.891em, 1000.33em, 4.336em, -999.998em); top: -5.263em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-113\"><span class=\"mrow\" id=\"MathJax-Span-114\"><span class=\"munderover\" id=\"MathJax-Span-115\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.24em, 4.336em, -999.998em); top: -4.005em; left: 0.049em;\"><span class=\"texatom\" id=\"MathJax-Span-116\"><span class=\"mrow\" id=\"MathJax-Span-117\"><span class=\"mi\" id=\"MathJax-Span-118\" style=\"font-family: MathJax_Main;\">j<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.33em, 3.777em, -999.998em); top: -4.471em; left: 0em;\"><span class=\"mo\" id=\"MathJax-Span-119\" style=\"\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00af<\/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 style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.819em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-130\"><span class=\"mrow\" id=\"MathJax-Span-131\"><span class=\"mn\" id=\"MathJax-Span-132\" 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: -2.421em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-139\"><span class=\"mrow\" id=\"MathJax-Span-140\"><span class=\"mn\" id=\"MathJax-Span-141\" 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: 4.802em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1000.51em, 6.573em, -999.998em); top: -4.844em; left: 2.985em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.845em, 1000.51em, 4.15em, -999.998em); top: -5.263em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-120\"><span class=\"mrow\" id=\"MathJax-Span-121\"><span class=\"munderover\" id=\"MathJax-Span-122\"><span style=\"display: inline-block; position: relative; width: 0.515em; 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-123\"><span class=\"mrow\" id=\"MathJax-Span-124\"><span class=\"mi\" id=\"MathJax-Span-125\" style=\"font-family: MathJax_Main;\">k<\/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-126\" 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: 0.189em;\"><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 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 style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.819em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-133\"><span class=\"mrow\" id=\"MathJax-Span-134\"><span class=\"mn\" id=\"MathJax-Span-135\" 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: -2.421em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-142\"><span class=\"mrow\" id=\"MathJax-Span-143\"><span class=\"mn\" id=\"MathJax-Span-144\" 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: 4.849em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-145\" style=\"vertical-align: 2.286em;\"><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: -0.231em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.722em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -0.976em; 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: 4.569em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -2.105em; border-left: 0px solid; width: 0px; height: 4.814em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-9\">\\left|\\begin{array}{lll}  \\overline{\\mathrm{i}} & \\overline{\\mathrm{j}} & \\overline{\\mathrm{k}} \\\\  1 & 1 & 1 \\\\  2 & 1 & 3  \\end{array}\\right|<\/script><br> = i\u0305(3 \u2013 1) \u2013 j\u0305(3 \u2013 2) + k\u0305(1 \u2013 2)<br> = 2i\u0305 \u2013 j\u0305 \u2013 k\u0305<br> |a\u0305 \u00d7 b\u0305| = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-10-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-146\" style=\"width: 8.763em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 7.551em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.354em, 1007.55em, 2.658em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-147\"><span class=\"msqrt\" id=\"MathJax-Span-148\"><span style=\"display: inline-block; position: relative; width: 4.849em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1003.92em, 4.243em, -999.998em); top: -4.005em; left: 0.841em;\"><span class=\"mrow\" id=\"MathJax-Span-149\"><span class=\"mn\" id=\"MathJax-Span-150\" style=\"font-family: MathJax_Main;\">4<\/span><span class=\"mo\" id=\"MathJax-Span-151\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-152\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-153\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-154\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">1<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.59em, 1003.92em, 3.917em, -999.998em); top: -4.518em; left: 0.841em;\"><span style=\"display: inline-block; position: relative; width: 3.917em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -4.005em; left: -0.091em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -4.005em; left: 3.264em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 0.375em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 0.888em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 1.354em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 1.866em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 2.332em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 2.845em;\">\u2212<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, 1000.84em, 4.336em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"font-family: MathJax_Main;\">\u221a<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-155\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"msqrt\" id=\"MathJax-Span-156\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.354em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0.841em;\"><span class=\"mrow\" id=\"MathJax-Span-157\"><span class=\"mn\" id=\"MathJax-Span-158\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.59em, 1000.51em, 3.917em, -999.998em); top: -4.564em; left: 0.841em;\"><span style=\"font-family: MathJax_Main;\">\u2013<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.078em, 1000.84em, 4.336em, -999.998em); top: -4.052em; left: 0em;\"><span style=\"font-family: MathJax_Main;\">\u221a<\/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: -0.268em; border-left: 0px solid; width: 0px; height: 1.3em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-10\">\\sqrt{4+1+1}=\\sqrt{6}<\/script><br> Unit vector perpendicular to both a\u0305 and b\u0305<br> = \u00b1<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-11-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-159\" style=\"width: 10.021em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 8.623em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1008.39em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-160\"><span class=\"mfrac\" id=\"MathJax-Span-161\"><span style=\"display: inline-block; position: relative; width: 1.82em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.124em, 1001.31em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.65em;\"><span class=\"mrow\" id=\"MathJax-Span-162\"><span class=\"munderover\" id=\"MathJax-Span-163\"><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-164\"><span class=\"mrow\" id=\"MathJax-Span-165\"><span class=\"mi\" id=\"MathJax-Span-166\" 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-167\" 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-168\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00d7<\/span><span class=\"munderover\" id=\"MathJax-Span-169\"><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-170\"><span class=\"mrow\" id=\"MathJax-Span-171\"><span class=\"mi\" id=\"MathJax-Span-172\" 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-173\" 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><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.124em, 1001.59em, 4.336em, -999.998em); top: -3.399em; left: 50%; margin-left: -0.836em;\"><span class=\"mrow\" id=\"MathJax-Span-174\"><span class=\"texatom\" id=\"MathJax-Span-175\"><span class=\"mrow\" id=\"MathJax-Span-176\"><span class=\"mo\" id=\"MathJax-Span-177\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">|<\/span><\/span><\/span><span class=\"munderover\" id=\"MathJax-Span-178\"><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-179\"><span class=\"mrow\" id=\"MathJax-Span-180\"><span class=\"mi\" id=\"MathJax-Span-181\" 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-182\" 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-183\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u00d7<\/span><span class=\"munderover\" id=\"MathJax-Span-184\"><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-185\"><span class=\"mrow\" id=\"MathJax-Span-186\"><span class=\"mi\" id=\"MathJax-Span-187\" 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-188\" 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=\"texatom\" id=\"MathJax-Span-189\"><span class=\"mrow\" id=\"MathJax-Span-190\"><span class=\"mo\" id=\"MathJax-Span-191\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">|<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.82em, 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: 1.82em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-192\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mo\" id=\"MathJax-Span-193\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">\u00b1<\/span><span class=\"mrow\" id=\"MathJax-Span-194\" style=\"padding-left: 0.189em;\"><span class=\"mo\" id=\"MathJax-Span-195\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-196\"><span style=\"display: inline-block; position: relative; width: 2.472em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.124em, 1002.33em, 4.289em, -999.998em); top: -4.564em; left: 50%; margin-left: -1.163em;\"><span class=\"mrow\" id=\"MathJax-Span-197\"><span class=\"mn\" id=\"MathJax-Span-198\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"munderover\" id=\"MathJax-Span-199\"><span style=\"display: inline-block; position: relative; width: 0.235em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.19em, 4.15em, -999.998em); top: -4.005em; left: 0.049em;\"><span class=\"texatom\" id=\"MathJax-Span-200\"><span class=\"mrow\" id=\"MathJax-Span-201\"><span class=\"mi\" id=\"MathJax-Span-202\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">i<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.497em, 1000.24em, 3.87em, -999.998em); top: -4.331em; left: 0em;\"><span class=\"mo\" id=\"MathJax-Span-203\" style=\"\"><span><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-204\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"munderover\" id=\"MathJax-Span-205\"><span style=\"display: inline-block; position: relative; width: 0.235em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.14em, 4.289em, -999.998em); top: -4.005em; left: 0.002em;\"><span class=\"texatom\" id=\"MathJax-Span-206\"><span class=\"mrow\" id=\"MathJax-Span-207\"><span class=\"mi\" id=\"MathJax-Span-208\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">j<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.497em, 1000.24em, 3.87em, -999.998em); top: -4.331em; left: 0em;\"><span class=\"mo\" id=\"MathJax-Span-209\" style=\"\"><span><span style=\"font-size: 50%; font-family: MathJax_Main;\">\u00af<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-210\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"munderover\" id=\"MathJax-Span-211\"><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-212\"><span class=\"mrow\" id=\"MathJax-Span-213\"><span class=\"mi\" id=\"MathJax-Span-214\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">k<\/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-215\" 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: 0.142em;\"><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><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.93em, 4.289em, -999.998em); top: -3.539em; left: 50%; margin-left: -0.464em;\"><span class=\"msqrt\" id=\"MathJax-Span-216\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-217\"><span class=\"mn\" id=\"MathJax-Span-218\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">6<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.38em, 1.26em, -999.998em); top: -1.675em; left: 0.608em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.044em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1000.61em, 4.289em, -999.998em); top: -4.005em; left: 0em;\"><span><span style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u221a<\/span><\/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 style=\"position: absolute; clip: rect(0.888em, 1002.47em, 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: 2.472em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-219\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">)<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.357em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.868em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-11\">\\frac{\\overline{\\mathrm{a}} \\times \\overline{\\mathrm{b}}}{|\\overline{\\mathrm{a}} \\times \\overline{\\mathrm{b}}|}=\\pm\\left(\\frac{2 \\overline{\\mathrm{i}}-\\overline{\\mathrm{j}}-\\overline{\\mathrm{k}}}{\\sqrt{6}}\\right)<\/script><\/p>\n\n<h3>Question 9.<br> If \u03b8 is the angle between the vectors i\u0305 + j\u0305 and j\u0305 + k\u0305, then find sin \u03b8.<\/h3>\n\n<p>Answer:<br> Let a\u0305 = i\u0305 + j\u0305 and b\u0305 = j\u0305 + k\u0305<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7097\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5b-2.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(b) 2\" width=\"232\" height=\"367\" sizes=\"auto, (max-width: 232px) 100vw, 232px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(b) 2\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(b)\"><\/p>\n\n<h3>Question 10.<br> Find the area of the parallelogram having a\u0305 = 2j\u0305 \u2013 k\u0305 and b\u0305 = \u2013 i\u0305 + k\u0305 as adjacent sides.<\/h3>\n\n<p>Answer:<br> Vector area of the parallelogram having<br> a\u0305 = 2j\u0305 \u2013 k\u0305 and b\u0305 = -i\u0305 + k\u0305 as adjacent sides = a\u0305 \u00d7 b\u0305<br> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-12-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-220\" 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, 6.293em, -999.998em); top: -4.471em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-221\"><span class=\"mrow\" id=\"MathJax-Span-222\"><span class=\"mo\" id=\"MathJax-Span-223\" style=\"vertical-align: 2.192em;\"><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: -0.37em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.281em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.349em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-224\"><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(2.425em, 1001.21em, 6.433em, -999.998em); top: -4.704em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1000.42em, 4.15em, -999.998em); top: -5.31em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-225\"><span class=\"mrow\" id=\"MathJax-Span-226\"><span class=\"texatom\" id=\"MathJax-Span-227\"><span class=\"mrow\" id=\"MathJax-Span-228\"><span class=\"munderover\" id=\"MathJax-Span-229\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.28em, 4.15em, -999.998em); top: -4.005em; left: 0.096em;\"><span class=\"mi\" id=\"MathJax-Span-230\" style=\"font-family: MathJax_Math-italic;\">i<\/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: -4.238em; left: 0em;\"><span class=\"mo\" id=\"MathJax-Span-231\" 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 style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.912em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-246\"><span class=\"mrow\" id=\"MathJax-Span-247\"><span class=\"mn\" id=\"MathJax-Span-248\" 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: -2.514em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-256\"><span class=\"mrow\" id=\"MathJax-Span-257\"><span class=\"mo\" id=\"MathJax-Span-258\" style=\"font-family: MathJax_Main;\">\u2212<\/span><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><span style=\"display: inline-block; width: 0px; height: 4.709em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1000.47em, 6.293em, -999.998em); top: -4.658em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1000.47em, 4.336em, -999.998em); top: -5.31em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-232\"><span class=\"mrow\" id=\"MathJax-Span-233\"><span class=\"texatom\" id=\"MathJax-Span-234\"><span class=\"mrow\" id=\"MathJax-Span-235\"><span class=\"munderover\" id=\"MathJax-Span-236\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.336em, -999.998em); top: -4.005em; left: 0.049em;\"><span class=\"mi\" id=\"MathJax-Span-237\" style=\"font-family: MathJax_Math-italic;\">j<\/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: -4.238em; left: 0em;\"><span class=\"mo\" id=\"MathJax-Span-238\" 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 style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.912em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-249\"><span class=\"mrow\" id=\"MathJax-Span-250\"><span class=\"mn\" id=\"MathJax-Span-251\" 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: -2.514em; 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;\">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: 4.662em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.293em, -999.998em); top: -4.658em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1000.51em, 4.15em, -999.998em); top: -5.31em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-239\"><span class=\"mrow\" id=\"MathJax-Span-240\"><span class=\"texatom\" id=\"MathJax-Span-241\"><span class=\"mrow\" id=\"MathJax-Span-242\"><span class=\"munderover\" id=\"MathJax-Span-243\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-244\" style=\"font-family: MathJax_Math-italic;\">k<\/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: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-245\" 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 style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.912em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-252\"><span class=\"mrow\" id=\"MathJax-Span-253\"><span class=\"mo\" id=\"MathJax-Span-254\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-255\" 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: -2.514em; 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><span style=\"display: inline-block; width: 0px; height: 4.662em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-266\" style=\"vertical-align: 2.192em;\"><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: -0.37em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.281em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.349em; 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: -1.997em; border-left: 0px solid; width: 0px; height: 4.597em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-12\">\\left|\\begin{array}{rrr}  \\bar{i} & \\bar{j} & \\bar{k} \\\\  0 & 2 & -1 \\\\  -1 & 0 & 1  \\end{array}\\right|<\/script> = 2 i\u0305 + j\u0305 + 2k\u0305<br> Area of the parallelogram<br> = |a\u0305 \u00d7 b\u0305| = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-13-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-267\" style=\"width: 5.641em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.849em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.354em, 1004.8em, 2.612em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-268\"><span class=\"msqrt\" id=\"MathJax-Span-269\"><span style=\"display: inline-block; position: relative; width: 4.849em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1003.96em, 4.243em, -999.998em); top: -4.005em; left: 0.841em;\"><span class=\"mrow\" id=\"MathJax-Span-270\"><span class=\"mn\" id=\"MathJax-Span-271\" style=\"font-family: MathJax_Main;\">4<\/span><span class=\"mo\" id=\"MathJax-Span-272\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-273\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-274\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-275\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">4<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.59em, 1003.92em, 3.917em, -999.998em); top: -4.518em; left: 0.841em;\"><span style=\"display: inline-block; position: relative; width: 3.917em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -4.005em; left: -0.091em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -4.005em; left: 3.264em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 0.375em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 0.888em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 1.354em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 1.866em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 2.332em;\">\u2212<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -4.005em; left: 2.845em;\">\u2212<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, 1000.84em, 4.336em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"font-family: MathJax_Main;\">\u221a<\/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: -0.268em; border-left: 0px solid; width: 0px; height: 1.246em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-13\">\\sqrt{4+1+4}<\/script> = 3 sq. units.<\/p>\n\n<h3>Question 11.<br> Find the area of the parallelogram whose diagonals are 3i\u0305 + j\u0305 \u2013 2k\u0305 and i\u0305 \u2013 3j\u0305 + 4k\u0305.<\/h3>\n","protected":false},"excerpt":{"rendered":"<p>I. Question 1. If |p\u0305| = 2, |q\u0305| = 3 and (p, q) = &#960;\/6 , then find |p\u0305 \u00d7 q\u0305|2. Answer: p\u0305 \u00d7 q\u0305 = |p\u0305| |q\u0305| sin\u03b8n\u0302 Given p\u0305 = 2, q\u0305 = 3 and (p\u0305. q\u0305) = &#960;\/6 |p\u0305 \u00d7 q\u0305| = (2) (3)sin &#960;\/6 =3 \u2234 |p\u0305 \u00d7 q\u0305|2 = 9 [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":2976,"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-2978","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(b)<\/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-5b-2\/\" \/>\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(b)\" \/>\r\n<meta property=\"og:description\" content=\"I. Question 1. If |p\u0305| = 2, |q\u0305| = 3 and (p, q) = &#960;\/6 , then find |p\u0305 \u00d7 q\u0305|2. Answer: p\u0305 \u00d7 q\u0305 = |p\u0305| |q\u0305| sin\u03b8n\u0302 Given p\u0305 = 2, q\u0305 = 3 and (p\u0305. q\u0305) = &#960;\/6 |p\u0305 \u00d7 q\u0305| = (2) (3)sin &#960;\/6 =3 \u2234 |p\u0305 \u00d7 q\u0305|2 = 9 [&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-5b-2\/\" \/>\r\n<meta property=\"og:site_name\" content=\"Manabadi Boards\" \/>\r\n<meta property=\"article:published_time\" content=\"2026-01-20T06:53:51+00:00\" \/>\r\n<meta property=\"article:modified_time\" content=\"2026-01-20T12:40:57+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-5b.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\" 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