{"id":2923,"date":"2026-01-19T15:49:39","date_gmt":"2026-01-19T10:19:39","guid":{"rendered":"https:\/\/www.manabadi.co.in\/boards\/?p=2923"},"modified":"2026-01-19T15:49:42","modified_gmt":"2026-01-19T10:19:42","slug":"ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a","status":"publish","type":"post","link":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\/","title":{"rendered":"TS Inter 1st Year Maths 1A Products of Vectors Solutions Exercise 5(a)"},"content":{"rendered":"\n<h3>Question 1.<br> Find the angle between the vectors i\u0305 + 2j\u0305 + 3k\u0305 and 3i\u0305 \u2013 j\u0305 + 2k\u0305. (Mar. \u201914)<\/h3>\n\n<p>Answer:<br> Let a\u0305 = i\u0305 + 2j\u0305 + 3k\u0305 and b\u0305 = 3i\u0305 \u2013 j\u0305 + 2k\u0305 and \u03b8 be the angle between them. Then<br> <img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-7087\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5a-1.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 1\" width=\"312\" height=\"194\" sizes=\"(max-width: 312px) 100vw, 312px\" data-pin-nopin=\"true\"><\/p>\n\n<h3>Question 2.<br> If the vectors 2i\u0305 + \u03bb j\u0305 \u2013 k\u0305 and 4i\u0305 \u2013 2j\u0305+ 2k\u0305 are perpendicular to each other then find \u03bb. [March, May 2005]<\/h3>\n\n<p>Answer:<br> Let a\u0305 = 2i\u0305 + \u03bb j\u0305 \u2013 k\u0305 and b\u0305 = 4i\u0305 \u2013 2j\u0305+ 2k\u0305 and If a\u0305 is perpendicular to b\u0305 then a\u0305.b\u0305 = o<br> \u21d2 (2i\u0305 + \u03bb j\u0305 \u2013 k\u0305).(4i\u0305 \u2013 2j\u0305+ 2k\u0305) = o<br> \u21d2 8 \u2013 2\u03bb \u2013 2 = 0 \u21d2 6 \u2013 2\u03bb = 0 \u21d2 \u03bb = 3<\/p>\n\n<h3>Question 3.<br> For what values of , the vectors i\u0305 \u2013 j\u0305 + 2k\u0305 and 8i\u0305 + 6j\u0305 \u2013 k\u0305 are at right angles?<\/h3>\n\n<p>Answer:<br> Let a\u0305 = i\u0305 \u2013 j\u0305 + 2k\u0305 and b\u0305 = 8i\u0305 + 6j\u0305 \u2013 k\u0305<br> If a\u0305, b\u0305 are right angles then a\u0305.b\u0305 = o<br> \u21d2 8 \u2013 6\u03bb \u2013 2 = 0<br> \u21d2 -6\u03bb + 6 = 0<br> \u21d2 \u03bb = 1<\/p>\n\n<h3>Question 4.<br> a\u0305 = 2i\u0305 \u2013 j\u0305 + k\u0305, b\u0305 = i\u0305 \u2013 3j\u0305 \u2013 5k\u0305. Find the vector c such that a, b and c form the sides of a triangle.<\/h3>\n\n<p>Answer:<br> a\u0305 = 2i\u0305 \u2013 j\u0305 + k\u0305, b\u0305 = i\u0305 \u2013 3j\u0305 \u2013 5k\u0305<br> \u2235 a\u0305, b\u0305, c\u0305 form the sides of a triangle a\u0305 + b\u0305 + c\u0305 = 0<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7086\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5a-2.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 2\" width=\"280\" height=\"182\" data-pin-nopin=\"true\"><br> \u2234 c\u0305 = -a\u0305 \u2013 b\u0305<br> = -(2i\u0305 \u2013 j\u0305 + k\u0305) \u2013 (i\u0305 \u2013 3 j\u0305 \u2013 5k\u0305)<br> = -3i\u0305 + 4j\u0305 + 4k\u0305<\/p>\n\n<h3>Question 5.<br> Find the angle between the planes r\u0305 . (2i\u0305 \u2013 j\u0305 + 2k\u0305) = 3 and r\u0305 .(3i\u0305 + 6j\u0305 + k\u0305) =4 (March 2015-T.S)<\/h3>\n\n<p>Answer:<br> If the angle between planes<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7085\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5a-3.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 3\" width=\"313\" height=\"205\" sizes=\"auto, (max-width: 313px) 100vw, 313px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 3\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a)\"><\/p>\n\n<h3>Question 6.<br> Let <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: 1.027em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.354em, 1000.89em, 2.565em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2\"><span class=\"msubsup\" id=\"MathJax-Span-3\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.078em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"munderover\" id=\"MathJax-Span-4\"><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-5\"><span class=\"mrow\" id=\"MathJax-Span-6\"><span class=\"mi\" id=\"MathJax-Span-7\" style=\"font-family: MathJax_Main;\">e<\/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-8\" 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 style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.468em;\"><span class=\"mn\" id=\"MathJax-Span-9\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/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: -0.214em; border-left: 0px solid; width: 0px; height: 1.192em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-1\">\\overline{\\mathrm{e}}_1<\/script> and <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-2-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-10\" style=\"width: 1.074em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.354em, 1000.93em, 2.565em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-11\"><span class=\"msubsup\" id=\"MathJax-Span-12\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.078em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"munderover\" id=\"MathJax-Span-13\"><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-14\"><span class=\"mrow\" id=\"MathJax-Span-15\"><span class=\"mi\" id=\"MathJax-Span-16\" style=\"font-family: MathJax_Main;\">e<\/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-17\" 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 style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.468em;\"><span class=\"texatom\" id=\"MathJax-Span-18\"><span class=\"mrow\" id=\"MathJax-Span-19\"><span class=\"texatom\" id=\"MathJax-Span-20\"><span class=\"mrow\" id=\"MathJax-Span-21\"><span class=\"mn\" id=\"MathJax-Span-22\" style=\"font-size: 70.7%; font-family: MathJax_Main-bold;\">2<\/span><\/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: -0.214em; border-left: 0px solid; width: 0px; height: 1.192em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-2\">\\overline{\\mathrm{e}}_{\\mathbf{2}}<\/script> be unit vectors making angle \u03b8. If <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-3-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-23\" style=\"width: 5.035em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.336em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.307em, 1004.2em, 2.845em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-24\"><span class=\"mfrac\" id=\"MathJax-Span-25\"><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-26\" 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-27\" 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 class=\"mrow\" id=\"MathJax-Span-28\"><span class=\"mo\" id=\"MathJax-Span-29\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Main;\">|<\/span><\/span><span class=\"msubsup\" id=\"MathJax-Span-30\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.078em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"munderover\" id=\"MathJax-Span-31\"><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-32\"><span class=\"mrow\" id=\"MathJax-Span-33\"><span class=\"mi\" id=\"MathJax-Span-34\" style=\"font-family: MathJax_Main;\">e<\/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-35\" 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 style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.468em;\"><span class=\"mn\" id=\"MathJax-Span-36\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-37\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"msubsup\" id=\"MathJax-Span-38\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.078em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"munderover\" id=\"MathJax-Span-39\"><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-40\"><span class=\"mrow\" id=\"MathJax-Span-41\"><span class=\"mi\" id=\"MathJax-Span-42\" style=\"font-family: MathJax_Main;\">e<\/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-43\" 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 style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.468em;\"><span class=\"mn\" id=\"MathJax-Span-44\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-45\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Main;\">|<\/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: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-3\">\\frac{1}{2}\\left|\\overline{\\mathrm{e}}_1-\\overline{\\mathrm{e}}_2\\right|<\/script> = sin \u03bb\u03b8, then find \u03bb.<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7084\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5a-4.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 4\" width=\"313\" height=\"490\" sizes=\"auto, (max-width: 313px) 100vw, 313px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 4\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a)\"><\/p>\n\n<p>Question 7.<br> Let a\u0305 = i\u0305 + j\u0305 + k\u0305 and b\u0305 = 2 i\u0305 + 3j\u0305 + k\u0305. Find<br> (i) the projection vector of bona and its magnitude<br> (ii) The vector components of b\u0305 in the direction of a\u0305 and perpendicular to a\u0305. [May 2006]<\/h3>\n\n<p>Answer:<br> Orthogonal projection of a vector b\u0305 on a\u0305 is<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7083\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5a-5.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 5\" width=\"315\" height=\"301\" sizes=\"auto, (max-width: 315px) 100vw, 315px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 5\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a)\"><\/p>\n\n<p>(ii) The component vector b in the direction of \u2013<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7082\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5a-6.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 6\" width=\"324\" height=\"293\" sizes=\"auto, (max-width: 324px) 100vw, 324px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 6\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a)\"><\/p>\n\n<h3>Question 8.<br> Find the equation of the plane through the point (3, \u2013 2, 1) and perpendicular to the vector (4, 7, \u2013 4).<\/h3>\n\n<p>Answer:<br> The equation of the plane passing through a\u0305 and perpendicular to the vector n\u0305 is r\u0305. n\u0305 = a\u0305. n\u0305<br> Given n\u0305 = 4i\u0305 + 7j\u0305 \u2013 4k\u0305 and a\u0305 = 3i\u0305 \u2013 2j\u0305 + k\u0305<br> r\u0305 . (4i\u0305 + 7j\u0305 \u2013 4k\u0305) \u2013 (3i\u0305 \u2013 2j\u0305 + k\u0305) . (4i\u0305 + 7j\u0305 \u2013 4k\u0305)<br> r . (4i\u0305 + 7j\u0305 \u2013 4k\u0305) = 12 \u2013 14 \u2013 4 = \u2013 6<br> \u21d2 r\u0305 . (-4i\u0305 \u2013 7j\u0305 + 4k\u0305) = 6<\/p>\n\n<h3>Question 9.<br> If a\u0305 = 2i\u0305 + 2j\u0305 \u2013 3k\u0305, b = 3i\u0305 \u2013 j\u0305 + 2k\u0305, then find the euigle between 2a\u0305 + b\u0305 and a\u0305 + 2b\u0305.<\/h3>\n\n<p>Answer:<br> Given a\u0305 = 2i\u0305 + 2j\u0305 \u2013 3k\u0305 and b\u0305 = 3i\u0305 \u2013 j\u0305 + 2k\u0305<br> We have<br> 2a\u0305 + b = 4i + 4j\u0305 \u2013 6k\u0305 + 3i\u0305 \u2013 j\u0305 + 2k\u0305 = 7i\u0305 + 3j\u0305 \u2013 4k\u0305<br> and a\u0305 + 2b\u0305 = (2i\u0305 + 2 j\u0305 \u2013 3k\u0305) + 2(31-7 + 2k) = 8i\u0305 + k\u0305<br> Let \u2018\u03b8\u2019 be the angle between the vectors 2a\u0305 + b\u0305 and a\u0305 + 2b\u0305<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7081\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5a-7.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 7\" width=\"289\" height=\"269\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 7\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a)\"><\/p>\n\n<h3>II.<br> Question 1.<br> Find the unit vector parallel to the XOY plane and perpendicular to the vector 4i\u0305 \u2013 3j\u0305 + k\u0305.<\/h3>\n\n<p>Answer:<br> Any vector parallel to XOY plane will be of the form xi\u0305 + yj\u0305.<br> The vector parallel to the XOY plane and perpendicular to the vector 4i\u0305 \u2013 3j\u0305 + k\u0305 is 3i\u0305 + 4j\u0305<br> Its magnitudes |3i\u0305 + 4j\u0305| = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-4-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-46\" style=\"width: 4.196em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.59em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.354em, 1003.54em, 2.612em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-47\"><span class=\"msqrt\" id=\"MathJax-Span-48\"><span style=\"display: inline-block; position: relative; width: 3.59em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1002.71em, 4.243em, -999.998em); top: -4.005em; left: 0.841em;\"><span class=\"mrow\" id=\"MathJax-Span-49\"><span class=\"mn\" id=\"MathJax-Span-50\" style=\"font-family: MathJax_Main;\">9<\/span><span class=\"mo\" id=\"MathJax-Span-51\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-52\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">16<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.59em, 1002.71em, 3.917em, -999.998em); top: -4.518em; left: 0.841em;\"><span style=\"display: inline-block; position: relative; width: 2.705em; 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: 2.006em;\">\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: 0.422em;\">\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: 0.981em;\">\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: 1.54em;\">\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-4\">\\sqrt{9+16}<\/script> = 5<br> Unit vector parallel to XOY plane and perpendicular to the vector 4i\u0305 \u2013 3j\u0305 + k\u0305 is<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-53\" style=\"width: 12.118em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 10.44em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1010.25em, 3.963em, -999.998em); top: -3.166em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-54\"><span class=\"mo\" id=\"MathJax-Span-55\" style=\"font-family: MathJax_Main;\">\u00b1<\/span><span class=\"mrow\" id=\"MathJax-Span-56\" style=\"padding-left: 0.189em;\"><span class=\"mo\" id=\"MathJax-Span-57\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-58\"><span style=\"display: inline-block; position: relative; width: 2.332em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.73em, 4.289em, -999.998em); top: -4.564em; left: 50%; margin-left: -0.883em;\"><span class=\"mrow\" id=\"MathJax-Span-59\"><span class=\"mn\" id=\"MathJax-Span-60\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"munderover\" id=\"MathJax-Span-61\"><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-62\"><span class=\"mrow\" id=\"MathJax-Span-63\"><span class=\"mi\" id=\"MathJax-Span-64\" 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-65\" 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-66\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-67\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span class=\"munderover\" id=\"MathJax-Span-68\"><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-69\"><span class=\"mrow\" id=\"MathJax-Span-70\"><span class=\"mi\" id=\"MathJax-Span-71\" 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-72\" 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><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.311em, 1002.19em, 4.289em, -999.998em); top: -3.539em; left: 50%; margin-left: -1.116em;\"><span class=\"msqrt\" id=\"MathJax-Span-73\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.59em, 4.196em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-74\"><span class=\"mn\" id=\"MathJax-Span-75\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">9<\/span><span class=\"mo\" id=\"MathJax-Span-76\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-77\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">16<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1001.63em, 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: 1.633em; 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, 1002.33em, 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: 2.332em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-78\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">)<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-79\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mo\" id=\"MathJax-Span-80\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">\u00b1<\/span><span class=\"mrow\" id=\"MathJax-Span-81\" style=\"padding-left: 0.189em;\"><span class=\"mo\" id=\"MathJax-Span-82\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-83\"><span style=\"display: inline-block; position: relative; width: 1.866em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.73em, 4.289em, -999.998em); top: -4.564em; left: 50%; margin-left: -0.883em;\"><span class=\"mrow\" id=\"MathJax-Span-84\"><span class=\"mn\" id=\"MathJax-Span-85\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"munderover\" id=\"MathJax-Span-86\"><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-87\"><span class=\"mrow\" id=\"MathJax-Span-88\"><span class=\"mi\" id=\"MathJax-Span-89\" 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-90\" 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-91\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-92\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span class=\"munderover\" id=\"MathJax-Span-93\"><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-94\"><span class=\"mrow\" id=\"MathJax-Span-95\"><span class=\"mi\" id=\"MathJax-Span-96\" 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-97\" 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><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-98\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.87em, 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: 1.866em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-99\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">)<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.171em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.808em; border-left: 0px solid; width: 0px; height: 2.327em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-5\">\\pm\\left(\\frac{3 \\overline{\\mathrm{i}}+4 \\overline{\\mathrm{j}}}{\\sqrt{9+16}}\\right)=\\pm\\left(\\frac{3 \\overline{\\mathrm{i}}+4 \\overline{\\mathrm{j}}}{5}\\right)<\/script><\/p>\n\n<h3>Question 2.<br> If a\u0305 + b\u0305 + c\u0305 = 0, |a\u0305I|= 3, |b\u0305| = 5 and |c\u0305| = 5 then find the angle between a\u0305 and b\u0305.<\/h3>\n\n<p>Answer:<br> Given a\u0305 + b\u0305 + c\u0305 = 0<br> c\u0305 = -(a\u0305 + b\u0305)<br> \u21d2 |c\u0305|<sup>2<\/sup> = (a\u0305 + b\u0305)<sup>2<\/sup> = a\u0305<sup>2<\/sup> + b\u0305<sup>2<\/sup> + 2(a\u0305. b\u0305)<br> \u21d2 49 = 9 + 25 + 2( .6)<\/p>\n\n<h3>Question 3.<br> If |a\u0305| = 2, |b\u0305| = 3 and |c\u0305| = 4 juid each of a\u0305, b\u0305, c\u0305 is perpendicular to the sum of the other two vectors, then find the magnitude of a\u0305 + b\u0305 + c\u0305.<\/h3>\n\n<p>Answer:<br> Given |a\u0305| = 2, |b\u0305| = 3 and |c\u0305| = 4<br> Since each of a\u0305, b\u0305, c\u0305 is perpendicular to the sum of other two vectors i.e., a\u0305 is perpendicular to b\u0305 + c\u0305<br> a\u0305 . (b\u0305 + c\u0305) = 0 \u21d2 a\u0305 . b\u0305 + a\u0305 . c\u0305 = 0<br> Similarly<br> b\u0305.(c\u0305 + a\u0305) = 0 \u21d2 b\u0305.c\u0305 + b\u0305.a\u0305 = 0<br> and c-(a + b) = 0 \u21d2 c\u0305. a\u0305 + c\u0305. b\u0305 = 0 Adding we get<br> 2 [(a\u0305 . b\u0305) + (b\u0305 . c\u0305) + (c\u0305 . a\u0305)] = 0 \u2026\u2026.(1)<br> Also (a\u0305 + b\u0305 + c\u0305)<br> = |a\u0305|<sup>2<\/sup> + |b\u0305|<sup>2<\/sup> + |c\u0305|<sup>2<\/sup> + 2(a\u0305.b\u0305 + b\u0305.c\u0305 + c\u0305.a\u0305)<br> = 4 + 9 + 16 + 2(a\u0305.b\u0305 + b\u0305. c\u0305 + c\u0305.a\u0305)<br> = 4 + 9 + 16 + 2 (0) = 29<br> \u2234 |a\u0305 + b\u0305 + c\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-100\" style=\"width: 2.192em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.866em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.307em, 1001.87em, 2.565em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-101\"><span class=\"msqrt\" id=\"MathJax-Span-102\"><span style=\"display: inline-block; position: relative; width: 1.866em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.98em, 4.15em, -999.998em); top: -4.005em; left: 0.841em;\"><span class=\"mrow\" id=\"MathJax-Span-103\"><span class=\"mn\" id=\"MathJax-Span-104\" style=\"font-family: MathJax_Main;\">29<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.59em, 1001.03em, 3.917em, -999.998em); top: -4.564em; left: 0.841em;\"><span style=\"display: inline-block; position: relative; width: 1.027em; 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: 0.329em;\">\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: -4.052em; 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: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.214em; border-left: 0px solid; width: 0px; height: 1.246em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-6\">\\sqrt{29}<\/script><\/p>\n\n<h3>Question 4.<br> Find the equation of the plane passing through the point a\u0305 = 2i\u0305 + 3j\u0305 \u2013 k\u0305 and perpendicular to the vector 3i\u0305 \u2013 2j\u0305 \u2013 2k\u0305 and the distance of this plane from the origin.<\/h3>\n\n<p>Answer:<br> Equation of the plane passing through the point a, and perpendicular to the vector n\u0305 is (r\u0305 \u2013 a\u0305) . n\u0305 = 0<br> \u21d2 7 . n\u0305 = a\u0305 . n\u0305<br> (liven a\u0305 = 2i\u0305 + 3 j\u0305 \u2013 k\u0305 and n\u0305 = 3i\u0305 \u2013 2j\u0305 \u2013 2k\u0305<br> We have r\u0305 . (3 i\u0305 \u2013 2 j\u0305 \u2013 2k\u0305)<br> = (2i\u0305 + 3j\u0305 \u2013 k\u0305) . (3i\u0305 \u2013 2j\u0305 \u2013 2k\u0305)<br> = 6 \u2013 6 + 2 = 2<br> \u21d2 r\u0305 . (3i\u0305 \u2013 2j\u0305 \u2013 2k\u0305) = 2<br> The distance from origin to this plane is<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7080\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5a-8.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 8\" width=\"237\" height=\"193\" data-pin-nopin=\"true\"><\/p>\n\n<h3>Question 5.<br> a\u0305, b\u0305, c\u0305 and d\u0305 are the position vectors of four coplanar points such that (a\u0305 \u2013 d\u0305) . (b\u0305 \u2013 c\u0305) = (b\u0305 \u2013 d\u0305) . (c\u0305 \u2013 a\u0305) = 0. Show that the point d represents the orthocentre of the triangle with a\u0305, b\u0305 and c\u0305 as its vertices.<\/h3>\n\n<p>Answer:<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7079\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5a-9.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 9\" width=\"176\" height=\"164\" data-pin-nopin=\"true\"><br>\n\nPosition vectors of A, B, C, D are a\u0305, b\u0305, c\u0305, d\u0305 respectively.<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-7-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-105\" 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-106\"><span class=\"munderover\" id=\"MathJax-Span-107\"><span style=\"display: inline-block; position: relative; width: 1.493em; 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-108\"><span class=\"mrow\" id=\"MathJax-Span-109\"><span class=\"mi\" id=\"MathJax-Span-110\" style=\"font-family: MathJax_Main;\">D<\/span><span class=\"mi\" id=\"MathJax-Span-111\" 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-112\" 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-113\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-114\" 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-115\"><span class=\"mrow\" id=\"MathJax-Span-116\"><span class=\"mi\" id=\"MathJax-Span-117\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-118\" 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-119\" 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 class=\"mo\" id=\"MathJax-Span-120\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"munderover\" id=\"MathJax-Span-121\" 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-122\"><span class=\"mrow\" id=\"MathJax-Span-123\"><span class=\"mi\" id=\"MathJax-Span-124\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-125\" 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-126\" 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: 2.332em;\"><\/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-7\">\\overline{\\mathrm{DA}}=\\overline{\\mathrm{OA}}-\\overline{\\mathrm{OD}}<\/script> = a\u0305 \u2013 d\u0305<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-8-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-127\" style=\"width: 8.157em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 7.039em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.167em, 1007.04em, 2.565em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-128\"><span class=\"munderover\" id=\"MathJax-Span-129\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.35em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-130\"><span class=\"mrow\" id=\"MathJax-Span-131\"><span class=\"mi\" id=\"MathJax-Span-132\" style=\"font-family: MathJax_Main;\">C<\/span><span class=\"mi\" id=\"MathJax-Span-133\" 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, 1001.4em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-134\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.4em; 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: 0.934em;\"><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-135\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-136\" 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-137\"><span class=\"mrow\" id=\"MathJax-Span-138\"><span class=\"mi\" id=\"MathJax-Span-139\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-140\" 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-141\" 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-142\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"munderover\" id=\"MathJax-Span-143\" style=\"padding-left: 0.235em;\"><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-144\"><span class=\"mrow\" id=\"MathJax-Span-145\"><span class=\"mi\" id=\"MathJax-Span-146\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-147\" 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-148\" 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: 2.332em;\"><\/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-8\">\\overline{\\mathrm{CB}}=\\overline{\\mathrm{OB}}-\\overline{\\mathrm{OC}}<\/script> = b\u0305 \u2013 c\u0305<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-9-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-149\" 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(1.167em, 1007.13em, 2.565em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-150\"><span class=\"munderover\" id=\"MathJax-Span-151\"><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-152\"><span class=\"mrow\" id=\"MathJax-Span-153\"><span class=\"mi\" id=\"MathJax-Span-154\" style=\"font-family: MathJax_Main;\">D<\/span><span class=\"mi\" id=\"MathJax-Span-155\" 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-156\" 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-157\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-158\" 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-159\"><span class=\"mrow\" id=\"MathJax-Span-160\"><span class=\"mi\" id=\"MathJax-Span-161\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-162\" 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-163\" 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-164\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"munderover\" id=\"MathJax-Span-165\" 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-166\"><span class=\"mrow\" id=\"MathJax-Span-167\"><span class=\"mi\" id=\"MathJax-Span-168\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-169\" 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-170\" 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: 2.332em;\"><\/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-9\">\\overline{\\mathrm{DB}}=\\overline{\\mathrm{OB}}-\\overline{\\mathrm{OD}}<\/script> = b\u0305 \u2013 d\u0305<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-10-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-171\" 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-172\"><span class=\"munderover\" id=\"MathJax-Span-173\"><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-174\"><span class=\"mrow\" id=\"MathJax-Span-175\"><span class=\"mi\" id=\"MathJax-Span-176\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-177\" 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-178\" 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-179\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-180\" 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-181\"><span class=\"mrow\" id=\"MathJax-Span-182\"><span class=\"mi\" id=\"MathJax-Span-183\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-184\" 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-185\" 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-186\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"munderover\" id=\"MathJax-Span-187\" 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-188\"><span class=\"mrow\" id=\"MathJax-Span-189\"><span class=\"mi\" id=\"MathJax-Span-190\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-191\" 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-192\" 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-10\">\\overline{\\mathrm{AC}}=\\overline{\\mathrm{OC}}-\\overline{\\mathrm{OA}}<\/script> = c\u0305 \u2013 a\u0305<br> Given (a\u0305 \u2013 d\u0305) . (b\u0305 \u2013 c\u0305) = 0<br> \u21d2 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-11-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-193\" style=\"width: 4.289em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.684em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1003.68em, 2.472em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-194\"><span class=\"munderover\" id=\"MathJax-Span-195\"><span style=\"display: inline-block; position: relative; width: 1.493em; 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-196\"><span class=\"mrow\" id=\"MathJax-Span-197\"><span class=\"mi\" id=\"MathJax-Span-198\" style=\"font-family: MathJax_Main;\">D<\/span><span class=\"mi\" id=\"MathJax-Span-199\" 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-200\" 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-201\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u22c5<\/span><span class=\"munderover\" id=\"MathJax-Span-202\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.35em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-203\"><span class=\"mrow\" id=\"MathJax-Span-204\"><span class=\"mi\" id=\"MathJax-Span-205\" style=\"font-family: MathJax_Main;\">C<\/span><span class=\"mi\" id=\"MathJax-Span-206\" 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, 1001.4em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-207\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.4em; 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: 0.934em;\"><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: -0.051em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-11\">\\overline{\\mathrm{DA}} \\cdot \\overline{\\mathrm{CB}}<\/script> = 0<br> \u21d2 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-12-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-208\" 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(1.074em, 1001.49em, 2.425em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-209\"><span class=\"munderover\" id=\"MathJax-Span-210\"><span style=\"display: inline-block; position: relative; width: 1.493em; 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-211\"><span class=\"mrow\" id=\"MathJax-Span-212\"><span class=\"mi\" id=\"MathJax-Span-213\" style=\"font-family: MathJax_Main;\">D<\/span><span class=\"mi\" id=\"MathJax-Span-214\" 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-215\" 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-12\">\\overline{\\mathrm{DA}}<\/script> is perpendicular to <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-13-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-216\" style=\"width: 1.68em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1001.45em, 2.425em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-217\"><span class=\"munderover\" id=\"MathJax-Span-218\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.35em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-219\"><span class=\"mrow\" id=\"MathJax-Span-220\"><span class=\"mi\" id=\"MathJax-Span-221\" style=\"font-family: MathJax_Main;\">B<\/span><span class=\"mi\" id=\"MathJax-Span-222\" 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, 1001.4em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-223\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.4em; 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: 0.934em;\"><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-13\">\\overline{\\mathrm{BC}}<\/script><br> \u2234 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-14-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-224\" 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(1.074em, 1001.49em, 2.425em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-225\"><span class=\"munderover\" id=\"MathJax-Span-226\"><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-227\"><span class=\"mrow\" id=\"MathJax-Span-228\"><span class=\"mi\" id=\"MathJax-Span-229\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-230\" 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-231\" 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-14\">\\overline{\\mathrm{AD}}<\/script> is an altitude of \u0394ABC<br> and (b\u0305 \u2013 d\u0305) . (c\u0305 \u2013 a\u0305) = 0<br> \u21d2 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-15-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-232\" style=\"width: 4.336em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.73em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1003.73em, 2.472em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-233\"><span class=\"munderover\" id=\"MathJax-Span-234\"><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-235\"><span class=\"mrow\" id=\"MathJax-Span-236\"><span class=\"mi\" id=\"MathJax-Span-237\" style=\"font-family: MathJax_Main;\">D<\/span><span class=\"mi\" id=\"MathJax-Span-238\" 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-239\" 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-240\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u22c5<\/span><span class=\"munderover\" id=\"MathJax-Span-241\" style=\"padding-left: 0.235em;\"><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-242\"><span class=\"mrow\" id=\"MathJax-Span-243\"><span class=\"mi\" id=\"MathJax-Span-244\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-245\" 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-246\" 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 style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/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-15\">\\overline{\\mathrm{DB}} \\cdot \\overline{\\mathrm{AC}}<\/script> = 0<br> \u21d2 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-16-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-247\" 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-248\"><span class=\"munderover\" id=\"MathJax-Span-249\"><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-250\"><span class=\"mrow\" id=\"MathJax-Span-251\"><span class=\"mi\" id=\"MathJax-Span-252\" style=\"font-family: MathJax_Main;\">D<\/span><span class=\"mi\" id=\"MathJax-Span-253\" 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-254\" 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 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: 1.3em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-16\">\\overline{\\mathrm{DB}}<\/script> is perpendicular to <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-17-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-255\" 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(1.074em, 1001.49em, 2.425em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-256\"><span class=\"munderover\" id=\"MathJax-Span-257\"><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-258\"><span class=\"mrow\" id=\"MathJax-Span-259\"><span class=\"mi\" id=\"MathJax-Span-260\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-261\" 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-262\" 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 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{AC}}<\/script><br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-18-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-263\" 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-264\"><span class=\"munderover\" id=\"MathJax-Span-265\"><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-266\"><span class=\"mrow\" id=\"MathJax-Span-267\"><span class=\"mi\" id=\"MathJax-Span-268\" style=\"font-family: MathJax_Main;\">D<\/span><span class=\"mi\" id=\"MathJax-Span-269\" 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-270\" 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 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: 1.3em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-18\">\\overline{\\mathrm{DB}}<\/script> another altitude \u0394ABC<br> Altitudes AD and BD intersect at D<br> D(d) is the orthocentre of \u0394ABC.<\/p>\n\n<h3>III.<br> Question 1.<br> Show that the points (5, \u2013 1, 1), (7, \u2013 4, 7), (1,-6, 10) and (- 1, \u2013 3, 4) are the vertices of a rhombus. (March 2013)<\/h3>\n\n<p>Answer:<br> Let A (5,-1, 1), B (7,-4, 7), C (1,-6, 10) and D (- 1, \u2013 3, 4) are the given points.<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7078\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5a-10.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 10\" width=\"295\" height=\"495\" sizes=\"auto, (max-width: 295px) 100vw, 295px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 10\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a)\"><br> \u2234 AB = BC = CD = DA = 7 units and AC \u2260 BD<br> \u2234 A, B, C, D are the points which are the vertices of a rhombus.<\/p>\n\n<h3>Question 2.<br> Let a\u0305 = 4i\u0305 + 5j\u0305 \u2013 k\u0305, b\u0305 = i\u0305 \u2013 4j\u0305 + 5k\u0305 and c\u0305 = 3i\u0305 + j\u0305 \u2013 k\u0305. Find the vector which is perpendicular to both a and b and whose magnitude is twenty one times the magnitude of c\u0305.<\/h3>\n\n<p>Answer:<br> Given a\u0305 = 4 i\u0305 + 5 j\u0305 \u2013 k\u0305<br> b\u0305 = i\u0305 \u2013 4 j\u0305 + 5k\u0305<br> and c\u0305 = 3 i\u0305 + j\u0305 \u2013 k\u0305<br> Let r\u0305 = xi\u0305 + yj\u0305 + zk\u0305 be the vector which is perpendicular to both a and b.<br> Then r\u0305. a\u0305 = 0 and r\u0305.b\u0305 = 0<br> \u21d2 4x + 5y \u2013 z = 0 \u2026\u2026\u2026\u2026..(1)<br> and x \u2013 4y + 5z = 0 \u2026\u2026\u2026.(2)<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7077\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5a-11.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 11\" width=\"291\" height=\"157\" data-pin-nopin=\"true\">\n\n<br> \u21d2 x = \u03bb, y = -\u03bb, z = -\u03bb<br> \u2234 The vector which is perpendicular to both a\u0305 and b\u0305 is r\u0305 = \u03bb(i\u0305 \u2013 j\u0305 \u2013 k\u0305)<br> Magnitude of c = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-19-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-271\" style=\"width: 9.369em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 8.064em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.354em, 1008.06em, 2.658em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-272\"><span class=\"msqrt\" id=\"MathJax-Span-273\"><span style=\"display: inline-block; position: relative; width: 4.849em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1003.92em, 4.243em, -999.998em); top: -4.005em; left: 0.841em;\"><span class=\"mrow\" id=\"MathJax-Span-274\"><span class=\"mn\" id=\"MathJax-Span-275\" style=\"font-family: MathJax_Main;\">9<\/span><span class=\"mo\" id=\"MathJax-Span-276\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-277\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-278\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-279\" 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-280\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"msqrt\" id=\"MathJax-Span-281\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.866em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.93em, 4.15em, -999.998em); top: -4.005em; left: 0.841em;\"><span class=\"mrow\" id=\"MathJax-Span-282\"><span class=\"mn\" id=\"MathJax-Span-283\" style=\"font-family: MathJax_Main;\">11<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.59em, 1001.03em, 3.917em, -999.998em); top: -4.564em; left: 0.841em;\"><span style=\"display: inline-block; position: relative; width: 1.027em; 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: 0.329em;\">\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: -4.052em; 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-19\">\\sqrt{9+1+1}=\\sqrt{11}<\/script><br> \u2234 The vector which is perpendicular to both a\u0305 and b\u0305 whose magnitude is 21 times the<br> magnitude of c\u0305 is = \u00b1 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-20-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-284\" style=\"width: 5.967em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.128em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(0.934em, 1005.13em, 3.124em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-285\"><span class=\"mfrac\" id=\"MathJax-Span-286\"><span style=\"display: inline-block; position: relative; width: 4.849em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.218em, 1004.66em, 4.336em, -999.998em); top: -4.564em; left: 50%; margin-left: -2.374em;\"><span class=\"mrow\" id=\"MathJax-Span-287\"><span class=\"mn\" id=\"MathJax-Span-288\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">21<\/span><span class=\"msqrt\" id=\"MathJax-Span-289\"><span style=\"display: inline-block; position: relative; width: 1.307em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.65em, 4.15em, -999.998em); top: -4.005em; left: 0.608em;\"><span class=\"mrow\" id=\"MathJax-Span-290\"><span class=\"mn\" id=\"MathJax-Span-291\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">11<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.981em, 1000.7em, 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: 0.701em; 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 class=\"mo\" id=\"MathJax-Span-292\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">(<\/span><span class=\"texatom\" id=\"MathJax-Span-293\"><span class=\"mrow\" id=\"MathJax-Span-294\"><span class=\"munderover\" id=\"MathJax-Span-295\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.24em, 4.15em, -999.998em); top: -4.005em; left: 0.049em;\"><span class=\"mi\" id=\"MathJax-Span-296\" style=\"font-size: 70.7%; 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(3.451em, 1000.33em, 3.777em, -999.998em); top: -4.192em; left: 0em;\"><span class=\"mo\" id=\"MathJax-Span-297\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-298\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-299\"><span class=\"mrow\" id=\"MathJax-Span-300\"><span class=\"munderover\" id=\"MathJax-Span-301\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.289em, -999.998em); top: -4.005em; left: 0.049em;\"><span class=\"mi\" id=\"MathJax-Span-302\" style=\"font-size: 70.7%; 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(3.451em, 1000.33em, 3.777em, -999.998em); top: -4.192em; left: 0em;\"><span class=\"mo\" id=\"MathJax-Span-303\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-304\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-305\"><span class=\"mrow\" id=\"MathJax-Span-306\"><span class=\"munderover\" id=\"MathJax-Span-307\"><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=\"mi\" id=\"MathJax-Span-308\" style=\"font-size: 70.7%; 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(3.451em, 1000.33em, 3.777em, -999.998em); top: -4.238em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-309\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-310\" 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(3.218em, 1002.47em, 4.336em, -999.998em); top: -3.446em; left: 50%; margin-left: -1.302em;\"><span class=\"mrow\" id=\"MathJax-Span-311\"><span class=\"texatom\" id=\"MathJax-Span-312\"><span class=\"mrow\" id=\"MathJax-Span-313\"><span class=\"mo\" id=\"MathJax-Span-314\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">|<\/span><\/span><\/span><span class=\"texatom\" id=\"MathJax-Span-315\"><span class=\"mrow\" id=\"MathJax-Span-316\"><span class=\"munderover\" id=\"MathJax-Span-317\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.24em, 4.15em, -999.998em); top: -4.005em; left: 0.049em;\"><span class=\"mi\" id=\"MathJax-Span-318\" style=\"font-size: 70.7%; 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(3.451em, 1000.33em, 3.777em, -999.998em); top: -4.192em; left: 0em;\"><span class=\"mo\" id=\"MathJax-Span-319\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-320\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-321\"><span class=\"mrow\" id=\"MathJax-Span-322\"><span class=\"munderover\" id=\"MathJax-Span-323\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.289em, -999.998em); top: -4.005em; left: 0.049em;\"><span class=\"mi\" id=\"MathJax-Span-324\" style=\"font-size: 70.7%; 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(3.451em, 1000.33em, 3.777em, -999.998em); top: -4.192em; left: 0em;\"><span class=\"mo\" id=\"MathJax-Span-325\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-326\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-327\"><span class=\"mrow\" id=\"MathJax-Span-328\"><span class=\"munderover\" id=\"MathJax-Span-329\"><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=\"mi\" id=\"MathJax-Span-330\" style=\"font-size: 70.7%; 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(3.451em, 1000.33em, 3.777em, -999.998em); top: -4.238em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-331\" 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><\/span><span class=\"texatom\" id=\"MathJax-Span-332\"><span class=\"mrow\" id=\"MathJax-Span-333\"><span class=\"mo\" id=\"MathJax-Span-334\" 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, 1004.85em, 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: 4.849em; 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: -0.862em; border-left: 0px solid; width: 0px; height: 2.327em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-20\">\\frac{21 \\sqrt{11}(\\bar{i}-\\bar{j}-\\bar{k})}{|\\bar{i}-\\bar{j}-\\bar{k}|}<\/script><br> = \u00b1 7<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-21-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-335\" style=\"width: 2.192em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.866em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.307em, 1001.87em, 2.565em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-336\"><span class=\"msqrt\" id=\"MathJax-Span-337\"><span style=\"display: inline-block; position: relative; width: 1.866em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.98em, 4.15em, -999.998em); top: -4.005em; left: 0.841em;\"><span class=\"mrow\" id=\"MathJax-Span-338\"><span class=\"mn\" id=\"MathJax-Span-339\" style=\"font-family: MathJax_Main;\">33<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.59em, 1001.03em, 3.917em, -999.998em); top: -4.564em; left: 0.841em;\"><span style=\"display: inline-block; position: relative; width: 1.027em; 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: 0.329em;\">\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: -4.052em; 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: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.214em; border-left: 0px solid; width: 0px; height: 1.246em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-21\">\\sqrt{33}<\/script> (i\u0305 \u2013 j\u0305 \u2013 k\u0305)<\/p>\n\n<h3>Question 3.<br> G is the centroid of \u0394ABC and a\u0305, b\u0305, c\u0305 are the lengths of the sides <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-22-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-340\" style=\"width: 3.963em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.404em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1003.4em, 2.658em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-341\"><span class=\"munderover\" id=\"MathJax-Span-342\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.35em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-343\"><span class=\"mrow\" id=\"MathJax-Span-344\"><span class=\"mi\" id=\"MathJax-Span-345\" style=\"font-family: MathJax_Main;\">B<\/span><span class=\"mi\" id=\"MathJax-Span-346\" 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, 1001.4em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-347\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.4em; 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: 0.934em;\"><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-348\" style=\"font-family: MathJax_Main;\">,<\/span><span class=\"munderover\" id=\"MathJax-Span-349\" style=\"padding-left: 0.189em;\"><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-350\"><span class=\"mrow\" id=\"MathJax-Span-351\"><span class=\"mi\" id=\"MathJax-Span-352\" style=\"font-family: MathJax_Main;\">C<\/span><span class=\"mi\" id=\"MathJax-Span-353\" 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, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-354\" 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 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: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-22\">\\overline{\\mathrm{B C}}, \\overline{\\mathrm{C A}}<\/script> and <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-23-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-355\" style=\"width: 1.68em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.074em, 1001.45em, 2.425em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-356\"><span class=\"munderover\" id=\"MathJax-Span-357\"><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-358\"><span class=\"mrow\" id=\"MathJax-Span-359\"><span class=\"mi\" id=\"MathJax-Span-360\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-361\" 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-362\" 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 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-23\">\\overline{\\mathrm{AB}}<\/script> respectively. Prove that <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-24-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-363\" style=\"width: 24.886em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 21.438em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1021.44em, 3.963em, -999.998em); top: -3.166em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-364\"><span class=\"msubsup\" id=\"MathJax-Span-365\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.264em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-366\"><span class=\"mrow\" id=\"MathJax-Span-367\"><span class=\"munderover\" id=\"MathJax-Span-368\"><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-369\" 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-370\" style=\"font-family: MathJax_Main;\">\u00af<\/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; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-371\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-372\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"msubsup\" id=\"MathJax-Span-373\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-374\"><span class=\"mrow\" id=\"MathJax-Span-375\"><span class=\"munderover\" id=\"MathJax-Span-376\"><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-377\" 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-378\" style=\"font-family: MathJax_Main;\">\u00af<\/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; top: -4.611em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-379\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-380\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"msubsup\" id=\"MathJax-Span-381\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.264em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-382\"><span class=\"mrow\" id=\"MathJax-Span-383\"><span class=\"munderover\" id=\"MathJax-Span-384\"><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-385\" 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-386\" style=\"font-family: MathJax_Main;\">\u00af<\/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; top: -4.378em; left: 0.562em;\"><span class=\"mn\" id=\"MathJax-Span-387\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-388\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mn\" id=\"MathJax-Span-389\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">3<\/span><span class=\"mrow\" id=\"MathJax-Span-390\" style=\"padding-left: 0.189em;\"><span class=\"mo\" id=\"MathJax-Span-391\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">(<\/span><\/span><span class=\"msubsup\" id=\"MathJax-Span-392\"><span style=\"display: inline-block; position: relative; width: 1.959em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.798em, 1001.54em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"munderover\" id=\"MathJax-Span-393\"><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-394\"><span class=\"mrow\" id=\"MathJax-Span-395\"><span class=\"mi\" id=\"MathJax-Span-396\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-397\" 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-398\" 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 style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.797em; left: 1.54em;\"><span class=\"mn\" id=\"MathJax-Span-399\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-400\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"msubsup\" id=\"MathJax-Span-401\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 1.913em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.845em, 1001.49em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"munderover\" id=\"MathJax-Span-402\"><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-403\"><span class=\"mrow\" id=\"MathJax-Span-404\"><span class=\"mi\" id=\"MathJax-Span-405\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-406\" 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-407\" 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 style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.751em; left: 1.493em;\"><span class=\"mn\" id=\"MathJax-Span-408\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-409\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"msubsup\" id=\"MathJax-Span-410\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 1.913em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.845em, 1001.49em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"munderover\" id=\"MathJax-Span-411\"><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-412\"><span class=\"mrow\" id=\"MathJax-Span-413\"><span class=\"mi\" id=\"MathJax-Span-414\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-415\" 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-416\" 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 style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.751em; left: 1.493em;\"><span class=\"mn\" id=\"MathJax-Span-417\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-418\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">)<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-419\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-420\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">9<\/span><span class=\"mo\" id=\"MathJax-Span-421\" style=\"font-family: MathJax_Main;\">(<\/span><span class=\"munderover\" id=\"MathJax-Span-422\"><span style=\"display: inline-block; position: relative; width: 1.587em; 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-423\"><span class=\"mrow\" id=\"MathJax-Span-424\"><span class=\"mi\" id=\"MathJax-Span-425\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-426\" style=\"font-family: MathJax_Main;\">G<\/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-427\" 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: 1.26em;\"><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=\"msubsup\" id=\"MathJax-Span-428\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.124em, 1000.28em, 4.383em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mo\" id=\"MathJax-Span-429\" style=\"font-family: MathJax_Main;\">)<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.375em;\"><span class=\"mn\" id=\"MathJax-Span-430\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.171em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.808em; border-left: 0px solid; width: 0px; height: 2.327em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-24\">\\bar{a}^2+\\bar{b}^2+\\bar{c}^2=3\\left(\\overline{\\mathrm{OA}}^2+\\overline{\\mathrm{OB}}^2+\\overline{\\mathrm{OC}}^2\\right)-9(\\overline{\\mathrm{OG}})^2<\/script>. where \u2018O\u2019 is any point.<\/h3>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7076\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5a-12.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 12\" width=\"186\" height=\"155\" data-pin-nopin=\"true\"><br>\n\n<p>Answer:<br> Given that <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-25-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-431\" 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, 2.658em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-432\"><span class=\"munderover\" id=\"MathJax-Span-433\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.35em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-434\"><span class=\"mrow\" id=\"MathJax-Span-435\"><span class=\"mi\" id=\"MathJax-Span-436\" style=\"font-family: MathJax_Main;\">B<\/span><span class=\"mi\" id=\"MathJax-Span-437\" 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, 1001.4em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-438\" style=\"\"><span style=\"display: inline-block; position: relative; width: 1.4em; 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: 0.934em;\"><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-439\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-440\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-441\"><span class=\"mrow\" id=\"MathJax-Span-442\"><span class=\"mi\" id=\"MathJax-Span-443\" 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, 1000.47em, 3.777em, -999.998em); top: -4.285em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-444\" 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 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 class=\"mo\" id=\"MathJax-Span-445\" style=\"font-family: MathJax_Main;\">,<\/span><span class=\"munderover\" id=\"MathJax-Span-446\" style=\"padding-left: 0.189em;\"><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-447\"><span class=\"mrow\" id=\"MathJax-Span-448\"><span class=\"mi\" id=\"MathJax-Span-449\" style=\"font-family: MathJax_Main;\">C<\/span><span class=\"mi\" id=\"MathJax-Span-450\" 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, 1001.45em, 3.777em, -999.998em); top: -4.518em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-451\" 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-452\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-453\" style=\"padding-left: 0.282em;\"><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-454\"><span class=\"mrow\" id=\"MathJax-Span-455\"><span class=\"mi\" id=\"MathJax-Span-456\" 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-457\" 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 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: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-25\">\\overline{\\mathrm{BC}}=\\overline{\\mathrm{a}}, \\overline{\\mathrm{CA}}=\\overline{\\mathrm{b}}<\/script> and <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-26-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-458\" style=\"width: 3.777em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.264em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1003.26em, 2.472em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-459\"><span class=\"munderover\" id=\"MathJax-Span-460\"><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-461\"><span class=\"mrow\" id=\"MathJax-Span-462\"><span class=\"mi\" id=\"MathJax-Span-463\" style=\"font-family: MathJax_Main;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-464\" 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-465\" 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-466\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-467\" style=\"padding-left: 0.282em;\"><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-468\"><span class=\"mrow\" id=\"MathJax-Span-469\"><span class=\"mi\" id=\"MathJax-Span-470\" 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-471\" 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 style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/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-26\">\\overline{\\mathrm{AB}}=\\overline{\\mathrm{c}}<\/script>.<br> Let O\u2019 be the origin and let p.q.r be the position vectors of A, B, C then <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-27-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-472\" style=\"width: 4.01em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.451em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1003.45em, 2.658em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-473\"><span class=\"munderover\" id=\"MathJax-Span-474\"><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-475\"><span class=\"mrow\" id=\"MathJax-Span-476\"><span class=\"mi\" id=\"MathJax-Span-477\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-478\" 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-479\" 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 class=\"mo\" id=\"MathJax-Span-480\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-481\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.336em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-482\"><span class=\"mrow\" id=\"MathJax-Span-483\"><span class=\"mi\" id=\"MathJax-Span-484\" style=\"font-family: MathJax_Main;\">p<\/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-485\" 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 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: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-27\">\\overline{\\mathrm{OA}}=\\overline{\\mathrm{p}}<\/script>, <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-28-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-486\" style=\"width: 9.369em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 8.064em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.167em, 1008.02em, 2.658em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-487\"><span class=\"munderover\" id=\"MathJax-Span-488\"><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-489\"><span class=\"mrow\" id=\"MathJax-Span-490\"><span class=\"mi\" id=\"MathJax-Span-491\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-492\" 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-493\" 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-494\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-495\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.336em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-496\"><span class=\"mrow\" id=\"MathJax-Span-497\"><span class=\"mi\" id=\"MathJax-Span-498\" style=\"font-family: MathJax_Main;\">q<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/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: 0.049em;\"><span class=\"mo\" id=\"MathJax-Span-499\" 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 class=\"mo\" id=\"MathJax-Span-500\" style=\"font-family: MathJax_Main;\">,<\/span><span class=\"mspace\" id=\"MathJax-Span-501\" style=\"height: 0em; vertical-align: 0em; width: 0.981em; display: inline-block; overflow: hidden;\"><\/span><span class=\"munderover\" id=\"MathJax-Span-502\" style=\"padding-left: 0.189em;\"><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-503\"><span class=\"mrow\" id=\"MathJax-Span-504\"><span class=\"mi\" id=\"MathJax-Span-505\" style=\"font-family: MathJax_Main;\">O<\/span><span class=\"mi\" id=\"MathJax-Span-506\" 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-507\" 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-508\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"munderover\" id=\"MathJax-Span-509\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.38em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-510\"><span class=\"mrow\" id=\"MathJax-Span-511\"><span class=\"mi\" id=\"MathJax-Span-512\" style=\"font-family: MathJax_Main;\">r<\/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: -4.285em; left: 0.002em;\"><span class=\"mo\" id=\"MathJax-Span-513\" 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 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: 1.516em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-28\">\\overline{\\mathrm{OB}}=\\overline{\\mathrm{q}}, \\quad \\overline{\\mathrm{OC}}=\\overline{\\mathrm{r}}<\/script> respectively.<br> Then the position vector of centroid<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7075\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5a-13.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 13\" width=\"365\" height=\"721\" sizes=\"auto, (max-width: 365px) 100vw, 365px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 13\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a)\"><\/p>\n\n<p>Question 4.<br> A line makes angles \u03b8<sub>1<\/sub>, \u03b8<sub>2<\/sub>, \u03b8<sub>3<\/sub>, and \u03b8<sub>4<\/sub> with the diagonals of a cube. Show that cos<sup>2<\/sup>\u03b8<sub>1<\/sub> + cos<sup>2<\/sup>\u03b8<sub>2<\/sub> + cos<sup>2<\/sup>\u03b8<sub>3<\/sub> + cos<sup>2<\/sup>\u03b8<sub>4<\/sub> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-29-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-514\" 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-515\"><span class=\"mfrac\" id=\"MathJax-Span-516\"><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-517\" 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, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-518\" 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-29\">\\frac{4}{3}<\/script>.<br>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7074\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5a-14.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 14\" width=\"171\" height=\"182\" data-pin-nopin=\"true\"><br> Let \u2018O\u2019 be the origin and \u2018a\u2019 be the length of the side of a cube.<br> i\u0305, j\u0305, k\u0305 are unit vectors along X, Y and Z axes respectively.<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7073\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5a-15.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 15\" width=\"326\" height=\"331\" sizes=\"auto, (max-width: 326px) 100vw, 326px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 15\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a)\"><br> Let r\u0305 = xi\u0305 + yj\u0305 + zk\u0305 be the line makes angles \u03b8<sub>1<\/sub>, \u03b8<sub>2<\/sub>, \u03b8<sub>3<\/sub>, \u03b8<sub>4<\/sub> with diagonals of a cube<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7072\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-5-Products-of-Vectors-Ex-5a-16.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 16\" width=\"319\" height=\"801\" sizes=\"auto, (max-width: 319px) 100vw, 319px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a) 16\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 5 Products of Vectors Ex 5(a)\"><\/p>\n\n\n","protected":false},"excerpt":{"rendered":"<p>Question 1. Find the angle between the vectors i\u0305 + 2j\u0305 + 3k\u0305 and 3i\u0305 \u2013 j\u0305 + 2k\u0305. (Mar. \u201914) Answer: Let a\u0305 = i\u0305 + 2j\u0305 + 3k\u0305 and b\u0305 = 3i\u0305 \u2013 j\u0305 + 2k\u0305 and \u03b8 be the angle between them. Then Question 2. If the vectors 2i\u0305 + \u03bb j\u0305 [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":2926,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1],"tags":[],"class_list":{"0":"post-2923","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-other"},"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(a)<\/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-5a\/\" \/>\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(a)\" \/>\r\n<meta property=\"og:description\" content=\"Question 1. Find the angle between the vectors i\u0305 + 2j\u0305 + 3k\u0305 and 3i\u0305 \u2013 j\u0305 + 2k\u0305. (Mar. \u201914) Answer: Let a\u0305 = i\u0305 + 2j\u0305 + 3k\u0305 and b\u0305 = 3i\u0305 \u2013 j\u0305 + 2k\u0305 and \u03b8 be the angle between them. Then Question 2. If the vectors 2i\u0305 + \u03bb j\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-5a\/\" \/>\r\n<meta property=\"og:site_name\" content=\"Manabadi Boards\" \/>\r\n<meta property=\"article:published_time\" content=\"2026-01-19T10:19:39+00:00\" \/>\r\n<meta property=\"article:modified_time\" content=\"2026-01-19T10:19:42+00:00\" \/>\r\n<meta property=\"og:image\" content=\"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a.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\" content=\"summary_large_image\" \/>\r\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Junaid\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"12 minutes\" \/>\r\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\\\/\"},\"author\":{\"name\":\"Junaid\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/#\\\/schema\\\/person\\\/a55695e135e9dc26536b290e31b4179d\"},\"headline\":\"TS Inter 1st Year Maths 1A Products of Vectors Solutions Exercise 5(a)\",\"datePublished\":\"2026-01-19T10:19:39+00:00\",\"dateModified\":\"2026-01-19T10:19:42+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\\\/\"},\"wordCount\":1218,\"commentCount\":0,\"image\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/boardscdn.manabadi.co.in\\\/wp-content\\\/uploads\\\/2026\\\/01\\\/07114619\\\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a.jpg\",\"articleSection\":[\"Other\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\\\/\",\"url\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\\\/\",\"name\":\"TS Inter 1st Year Maths 1A Products of Vectors Solutions Exercise 5(a)\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/boardscdn.manabadi.co.in\\\/wp-content\\\/uploads\\\/2026\\\/01\\\/07114619\\\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a.jpg\",\"datePublished\":\"2026-01-19T10:19:39+00:00\",\"dateModified\":\"2026-01-19T10:19:42+00:00\",\"author\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/#\\\/schema\\\/person\\\/a55695e135e9dc26536b290e31b4179d\"},\"breadcrumb\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\\\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\\\/#primaryimage\",\"url\":\"https:\\\/\\\/boardscdn.manabadi.co.in\\\/wp-content\\\/uploads\\\/2026\\\/01\\\/07114619\\\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a.jpg\",\"contentUrl\":\"https:\\\/\\\/boardscdn.manabadi.co.in\\\/wp-content\\\/uploads\\\/2026\\\/01\\\/07114619\\\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a.jpg\",\"width\":900,\"height\":600,\"caption\":\"TS Inter 1st Year Maths 1A Products of Vectors Solutions Exercise 5(a)\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"TS Inter 1st Year Maths 1A Products of Vectors Solutions Exercise 5(a)\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/#website\",\"url\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/\",\"name\":\"Manabadi Boards\",\"description\":\"\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/#\\\/schema\\\/person\\\/a55695e135e9dc26536b290e31b4179d\",\"name\":\"Junaid\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g\",\"url\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g\",\"contentUrl\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g\",\"caption\":\"Junaid\"}}]}<\/script>\r\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"TS Inter 1st Year Maths 1A Products of Vectors Solutions Exercise 5(a)","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\/","og_locale":"en_US","og_type":"article","og_title":"TS Inter 1st Year Maths 1A Products of Vectors Solutions Exercise 5(a)","og_description":"Question 1. Find the angle between the vectors i\u0305 + 2j\u0305 + 3k\u0305 and 3i\u0305 \u2013 j\u0305 + 2k\u0305. (Mar. \u201914) Answer: Let a\u0305 = i\u0305 + 2j\u0305 + 3k\u0305 and b\u0305 = 3i\u0305 \u2013 j\u0305 + 2k\u0305 and \u03b8 be the angle between them. Then Question 2. If the vectors 2i\u0305 + \u03bb j\u0305 [&hellip;]","og_url":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\/","og_site_name":"Manabadi Boards","article_published_time":"2026-01-19T10:19:39+00:00","article_modified_time":"2026-01-19T10:19:42+00:00","og_image":[{"width":900,"height":600,"url":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a.jpg","type":"image\/jpeg"}],"author":"Junaid","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Junaid","Est. reading time":"12 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\/#article","isPartOf":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\/"},"author":{"name":"Junaid","@id":"https:\/\/www.manabadi.co.in\/boards\/#\/schema\/person\/a55695e135e9dc26536b290e31b4179d"},"headline":"TS Inter 1st Year Maths 1A Products of Vectors Solutions Exercise 5(a)","datePublished":"2026-01-19T10:19:39+00:00","dateModified":"2026-01-19T10:19:42+00:00","mainEntityOfPage":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\/"},"wordCount":1218,"commentCount":0,"image":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\/#primaryimage"},"thumbnailUrl":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a.jpg","articleSection":["Other"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\/","url":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\/","name":"TS Inter 1st Year Maths 1A Products of Vectors Solutions Exercise 5(a)","isPartOf":{"@id":"https:\/\/www.manabadi.co.in\/boards\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\/#primaryimage"},"image":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\/#primaryimage"},"thumbnailUrl":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a.jpg","datePublished":"2026-01-19T10:19:39+00:00","dateModified":"2026-01-19T10:19:42+00:00","author":{"@id":"https:\/\/www.manabadi.co.in\/boards\/#\/schema\/person\/a55695e135e9dc26536b290e31b4179d"},"breadcrumb":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\/#primaryimage","url":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a.jpg","contentUrl":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a.jpg","width":900,"height":600,"caption":"TS Inter 1st Year Maths 1A Products of Vectors Solutions Exercise 5(a)"},{"@type":"BreadcrumbList","@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-products-of-vectors-solutions-exercise-5a\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/www.manabadi.co.in\/boards\/"},{"@type":"ListItem","position":2,"name":"TS Inter 1st Year Maths 1A Products of Vectors Solutions Exercise 5(a)"}]},{"@type":"WebSite","@id":"https:\/\/www.manabadi.co.in\/boards\/#website","url":"https:\/\/www.manabadi.co.in\/boards\/","name":"Manabadi Boards","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.manabadi.co.in\/boards\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/www.manabadi.co.in\/boards\/#\/schema\/person\/a55695e135e9dc26536b290e31b4179d","name":"Junaid","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/secure.gravatar.com\/avatar\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g","caption":"Junaid"}}]}},"_links":{"self":[{"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts\/2923","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/comments?post=2923"}],"version-history":[{"count":1,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts\/2923\/revisions"}],"predecessor-version":[{"id":2924,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts\/2923\/revisions\/2924"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/media\/2926"}],"wp:attachment":[{"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/media?parent=2923"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/categories?post=2923"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/tags?post=2923"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}