{"id":15051,"date":"2021-05-16T12:53:29","date_gmt":"2021-05-16T07:23:29","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=15051"},"modified":"2024-04-18T09:00:24","modified_gmt":"2024-04-18T03:30:24","slug":"n-2-2-product-of-two-vectors-exercise-problems-with-solutions","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=15051","title":{"rendered":"N \u2013 2.2 \u2013 Product of two vectors \u2013 Exercise Problems with solutions"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- A}}\\]<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {1\\ .}\\ \\color {red} { What\\ are\\ the\\ values\\ of}\\  \\overrightarrow{i}\\ . \\  \\overrightarrow{j}\\ ,\\ and\\ \\overrightarrow{k}\\ .\\ \\overrightarrow{k}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\   \\overrightarrow{i}\\ . \\  \\overrightarrow{j}\\ =\\ 0\\  and\\  \\overrightarrow{k}\\ .\\ \\overrightarrow{k}\\ =\\ 0\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {2\\ .}\\ \\color {red} {Find\\  the\\  scalar\\  product\\  of}\\  \\overrightarrow{i}+\\overrightarrow{j}\\  ,\\  \\overrightarrow{i}+\\overrightarrow{j}+ 3\\overrightarrow{k}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}= \\overrightarrow{i} + \\overrightarrow{j} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= \\overrightarrow{i}+\\overrightarrow{j}+ 3\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}.\\overrightarrow{b}= (\\overrightarrow{i}+\\overrightarrow{j}) .(\\overrightarrow{i}+\\overrightarrow{j}+ 3\\overrightarrow{k})\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\"> =  1(1)  + 1(1)  +  0 <\/p>\n\n\n\n<p class=\"has-text-align-center\">  =   1 + 1 <\/p>\n\n\n\n<p class=\"has-text-align-center\">  =   2 <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}.\\overrightarrow{b}= 2\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/3bm11HtAk5c\" title=\"Product of Vectors (Exercise) - Part - 2\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {3\\ .}\\ \\color {red} {Show\\ that\\ the\\ vectors}\\ \\overrightarrow{i} &#8211; 3\\overrightarrow{j} + 5\\overrightarrow{k}\\  and\\  &#8211; 2\\overrightarrow{i}+ 6\\overrightarrow{j}+4\\overrightarrow{k}\\ are\\ perpendicular\\ to\\ each\\ other\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}= \\overrightarrow{i} &#8211; 3\\overrightarrow{j} + 5\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= &#8211; 2\\overrightarrow{i}+ 6\\overrightarrow{j}+4\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}.\\overrightarrow{b}= (\\overrightarrow{i} &#8211; 3\\overrightarrow{j} + 5\\overrightarrow{k}) .(- 2\\overrightarrow{i}+ 6\\overrightarrow{j}+4\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 1(-2)\\ +\\ -3(6)\\ +\\ 5 (4)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  -\\ 2\\ -\\  18\\  +\\ 20\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"> \\[=\\   0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\\ and\\ \\overrightarrow{b}\\  are\\  perpendicular\\  vectors.\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/X-sFSLC6C3A\" title=\"Product of Vectors (Exercise) - Part - 3\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {4\\ .}\\ \\color {red} {Prove\\ that\\ the\\ two\\ vectors}\\ 8\\ \\overrightarrow{i}\\ +\\ 7\\overrightarrow{j}\\ -\\ \\overrightarrow{k}\\  and\\  3\\overrightarrow{i}\\ -\\ 3\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k}\\ are\\ perpendicular\\ to\\ each\\ other\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\\ =\\ 8\\ \\overrightarrow{i}\\ +\\ 7\\overrightarrow{j}\\ -\\ \\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}\\ =\\ 3\\overrightarrow{i}\\ -\\ 3\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}.\\overrightarrow{b}= (8\\ \\overrightarrow{i}\\ +\\ 7\\overrightarrow{j}\\ -\\ \\overrightarrow{k}) .(3\\overrightarrow{i}\\ -\\ 3\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 8(3)\\ +\\ 7(-3)\\ +\\ -1 (3)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  \\ 24\\ -\\  21\\  -\\ 3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"> \\[=\\   0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\overrightarrow{a}\\ and\\ \\overrightarrow{b}\\  are\\  perpendicular\\  vectors.\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/DQAmgkgSWrY\" title=\"Product of vectors (Exercise) - Part - 4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {5\\ .}\\ \\color {red} {Find\\ the\\ value\\ of\\ p}\\ if\\ the\\ vectors\\ 2\\overrightarrow{i}+ \\overrightarrow{j}- 5\\overrightarrow{k}\\  and\\  p\\overrightarrow{i}+ 3\\overrightarrow{j} &#8211; 2\\overrightarrow{k} are\\ perpendicular.\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}= 2\\overrightarrow{i}+ \\overrightarrow{j}- 5\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{b}= p\\overrightarrow{i}+ 3\\overrightarrow{j} &#8211; 2\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\  \\overrightarrow{a} and\\  \\overrightarrow{b} are\\ perpendicular\\ to\\ each\\ other \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[i.e\\  \\overrightarrow{a}.\\overrightarrow{b}= 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(2\\overrightarrow{i}+ \\overrightarrow{j}- 5\\overrightarrow{k} ).(p\\overrightarrow{i}+ 3\\overrightarrow{j} &#8211; 2\\overrightarrow{k}) = 0 .\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">    2(p) + 1(3) &#8211; 5 (-2)  =   0 <\/p>\n\n\n\n<p class=\"has-text-align-center\"> 2p + 3   +10  = 0<\/p>\n\n\n\n<p class=\"has-text-align-center\">  2p + 13   =  0  <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[p = \\frac{-13}{2}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/uf5SecquPIk\" title=\"Product of Vectors (Exercise) - Part - 04\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {6\\ .}\\ \\overrightarrow{a}\\ and\\ \\overrightarrow{b}\\ are\\ two\\ vectors\\ such\\ that\\ |\\overrightarrow{a}| = 6,\\ |\\overrightarrow{b}|= 4\\ and\\ \\overrightarrow{a}\\ . \\overrightarrow{b}\\ =12\\ \\hspace{5cm}\\]\\[\\color {red} {find\\ the\\ angle\\ between\\ them.}\\ \\hspace{3cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ |\\overrightarrow{a}| = 6,\\ |\\overrightarrow{b}|= 4\\ and\\ \\overrightarrow{a}\\ . \\overrightarrow{b}\\ =12\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\  cos\\  \\theta =\\frac{\\overrightarrow{a}.\\overrightarrow{b}}{\\overrightarrow{|a|}\\overrightarrow{|b|}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\frac{12}{(6)(4)} = \\frac{1}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\theta\\ =\\ cos^{1}(\\frac{1}{2})\\ =\\ 60^0\\ =\\ \\frac{\\pi}{3}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/QrjjlXWAs0E\" title=\"Product of Vectors (Exercise) - Part - 5\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {7\\ .}\\ \\color {red} {What\\ are\\ the\\ values\\ of}\\  (i)\\ \\overrightarrow{i}\\ . \\  \\overrightarrow{i}\\ ,\\ and\\ (ii)\\ \\overrightarrow{i}\\ \u00d7\\ \\overrightarrow{j}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\  (i)\\  \\overrightarrow{i}\\ . \\  \\overrightarrow{i}\\ =\\ i\\  \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\  \\overrightarrow{i}\\ \u00d7\\ \\overrightarrow{j}\\ =\\ \\overrightarrow{k}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {8\\ .}\\ \\color {red} {Prove\\ that\\ the\\ vectors}\\ \\overrightarrow{a}\\ =\\ 4\\overrightarrow{i}\\ -\\  2\\overrightarrow{j}\\ -\\ 6\\overrightarrow{k}\\ and\\ \\overrightarrow{b}\\ =\\  2\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ -\\ 3\\overrightarrow{k}\\ \\color {red} {are\\ parallel}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[To\\ show\\ \\overrightarrow{a}\u00d7\\overrightarrow{b} =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}= 4\\overrightarrow{i}\\ -\\  2\\overrightarrow{j}\\ -\\ 6\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= 2\\overrightarrow{i}-\\overrightarrow{j}\\ -\\ 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\u00d7\\overrightarrow{b} =\\begin{vmatrix}\n\\overrightarrow{i} &amp; \\overrightarrow{j} &amp; \\overrightarrow{k}\\\\\n4 &amp; -2 &amp; -6\\\\\n2 &amp; -1 &amp; -3\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( 6\\ -\\ 6)\\  -\\overrightarrow{j}(-12\\ +\\ 12)\\ +\\ \\overrightarrow{k}(-4\\ +\\ 4)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(0) -\\overrightarrow{j}(0)+\\overrightarrow{k}(0)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{a}\u00d7\\overrightarrow{b} =\\ 0}\\ \\hspace{7cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ The\\ given\\ vectors\\ \\overrightarrow{a}\\ and\\ \\overrightarrow{b}\\ are\\ parallel\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/x1bn4mEI0Lc\" title=\"Product of Vectors (Exercise) - Part - 07\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {9\\ .}\\ If\\ \\overrightarrow{a}\\ and\\ \\overrightarrow{b}\\ are\\ two\\ adjacent\\ sides\\ of\\ a\\ parallelogram.\\ \\hspace{15cm}\\]\\[\\color {red} {What\\ is\\ its\\ area?}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\  Area\\ of \\ parellelogram = |\\overrightarrow{a} \u00d7 \\overrightarrow{b}|\\  \\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {10\\ .}\\  If\\ |\\overrightarrow{a}| = 3,\\ |\\overrightarrow{b}|= 5\\ and\\ |\\overrightarrow{a} \u00d7 \\overrightarrow{b}|=10,\\ \\color {red} {find\\ the\\ angle\\ between\\ \\overrightarrow{a}\\ and\\ \\overrightarrow{b}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\  sin\\  \\theta =\\frac{|\\overrightarrow{a} \u00d7 \\overrightarrow{b}|}{\\overrightarrow{|a|}\\overrightarrow{|b|}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\frac{10}{(3)(5)} = \\frac{2}{3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\theta\\ = sin^{-1}(\\frac{2}{3})}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/5VICvoloYFM\" title=\"Product of Vectors (Exercise) - Part - 9\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- Ad1 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8240817448\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\underline{PART\\ -\\ B}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[11.\\ Find\\ the\\ projection\\ of\\ the\\ vector\\ 2\\overrightarrow{i}+ \\overrightarrow{j}- 2\\overrightarrow{k} on\\  the\\ vector\\  \\overrightarrow{i} &#8211; 2\\overrightarrow{j} &#8211; 2\\overrightarrow{k}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<p>Soln: <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}= 2\\overrightarrow{i}+ \\overrightarrow{j}- 2\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}=\\overrightarrow{i} &#8211; 2\\overrightarrow{j} &#8211; 2\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Projection\\  of\\ \\overrightarrow{a} on \\overrightarrow{b} =\\frac{\\overrightarrow{a}.\\overrightarrow{b}}{\\overrightarrow{|b|}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{(2\\overrightarrow{i}+ \\overrightarrow{j}- 2\\overrightarrow{k}).(\\overrightarrow{i} &#8211; 2\\overrightarrow{j} &#8211; 2\\overrightarrow{k})}{\\sqrt{(1)^2 + (-2)^2 + (-2)^2 }}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  \\frac{2(1)+ 1(-2)- 2(-2)}{\\sqrt{(1 + 4 + 4 }}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  \\frac{2 &#8211; 2 + 4}{\\sqrt{9}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Projection\\  of\\ \\overrightarrow{a} on \\overrightarrow{b} =\\frac{4}{3}}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/QLkuKBnHooQ\" title=\"Product of Vectors (Exercise) - Part - 11\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[12.\\ Find\\ the\\ projection\\ of\\ the\\ vector\\ 2\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\ \\overrightarrow{k}\\ on\\  the\\ vector\\  3\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ +\\  \\overrightarrow{k}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Soln:\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}= 2\\overrightarrow{i}+ \\overrightarrow{j}- 2\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}=\\overrightarrow{i} &#8211; 2\\overrightarrow{j} &#8211; 2\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Projection\\  of\\ \\overrightarrow{a} on \\overrightarrow{b} =\\frac{\\overrightarrow{a}.\\overrightarrow{b}}{\\overrightarrow{|b|}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{(2\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\ \\overrightarrow{k}).(3\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ +\\  \\overrightarrow{k})}{\\sqrt{(3)^2 + (-1)^2 + (1)^2 }}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  \\frac{2(3)\\ +\\ 3(-\\ 1)\\ +\\ 1(1)}{\\sqrt{(9\\ +\\  1\\ +\\  1 }}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  \\frac{6\\ -\\  3\\ +\\  1}{\\sqrt{11}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Projection\\  of\\ \\overrightarrow{a} on \\overrightarrow{b}\\ =\\ \\frac{4}{\\sqrt{11}}}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/yaugpzWBUKk\" title=\"Product of Vectors - Part - 12\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {13\\ .}\\ \\color {red} {Find\\ the\\ work\\ done}\\ by\\ the\\ force\\  \\overrightarrow{i}\\ -\\  7\\overrightarrow{j}\\ &#8211; 2\\overrightarrow{k},\\ \\hspace{8cm}\\]\\[when\\ the\\ displacement\\ is\\  3\\overrightarrow{i}\\ -\\ 5\\overrightarrow{j}\\  -\\ 4\\overrightarrow{k}\\ \\hspace{7cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F}\\ =\\ \\overrightarrow{i}\\ -\\ 7\\overrightarrow{j}\\ -\\ 2\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{d}\\ =\\ 3\\overrightarrow{i}\\ -\\ 5\\overrightarrow{j}\\ -\\ 4\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Work\\  done =  \\overrightarrow{F}.\\overrightarrow{d}= (\\overrightarrow{i}\\ -\\ 7\\overrightarrow{j}\\ -\\ 2\\overrightarrow{k}) .(3\\overrightarrow{i}\\ &#8211; 5\\overrightarrow{j}\\ -\\ 4\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= 1 ( 3 )\\ +\\ -7 ( &#8211; 5 )\\ +\\ -2 ( -4 )\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 3\\ +\\  35\\ +\\ 8\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Work\\ done\\  =\\  46\\ units}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/cg_80ccLZwY\" title=\"Product of vectors (Exercise) - Part - 13\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {14\\ .}\\ \\color {red} {Find\\  the\\ cross\\ product\\ of} \\  2\\overrightarrow{i}\\ -\\ 5\\ \\overrightarrow{j}\\ -\\  \\overrightarrow{k}\\ and\\ \\overrightarrow{i}\\ +\\ 6\\ \\overrightarrow{j}\\ -\\ \\overrightarrow{k}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\\ =\\ 2\\overrightarrow{i}\\ -\\ 5\\ \\overrightarrow{j}\\ -\\  \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}\\ =\\ \\overrightarrow{i}\\ +\\ 6\\ \\overrightarrow{j}\\ -\\ \\overrightarrow{k}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\u00d7\\overrightarrow{b} =\\begin{vmatrix}\n\\overrightarrow{i} &amp; \\overrightarrow{j} &amp; \\overrightarrow{k}\\\\\n2 &amp; &#8211; 5 &amp; &#8211; 1\\\\\n1 &amp; 6 &amp; &#8211; 1\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(5 + 6) -\\overrightarrow{j}(-\\ 2\\ +\\ 1)\\ +\\ \\overrightarrow{k}(12\\ +\\ 5)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(11) -\\overrightarrow{j}(-\\ 1)+\\overrightarrow{k}(17)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{ \\overrightarrow{a}\u00d7 \\overrightarrow{b}\\ =\\ 11\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ +\\ 17\\ \\overrightarrow{k}}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/X6PfkDXQPlQ\" title=\"Product of Vectors ( Exercise) -Part - 14\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[15.\\ Find\\ the\\ area\\ of\\ the\\ parellelogram\\ whose\\ adjacent\\ sides\\ are\\ \\overrightarrow{i} + \\overrightarrow{j} + 3\\overrightarrow{k}\\  and\\  2\\overrightarrow{i} + \\overrightarrow{j}+ 2\\overrightarrow{k}.\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Soln:\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}=\\overrightarrow{i} + \\overrightarrow{j} + 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}=  2\\overrightarrow{i} + \\overrightarrow{j}+ 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Area\\ of \\ parellelogram = |\\overrightarrow{a} \u00d7 \\overrightarrow{b}|\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\u00d7\\overrightarrow{b} =\\begin{vmatrix}\n\\overrightarrow{i} &amp;  \\overrightarrow{j} &amp; \\overrightarrow{k}\\\\\n1 &amp;  1 &amp; 3\\\\\n2 &amp; 1 &amp; 2\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( 2 -3) -\\overrightarrow{j}(3+1)+\\overrightarrow{k}(3-0)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(-1) -\\overrightarrow{j}(- 4)+\\overrightarrow{k}(- 1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}\u00d7 \\overrightarrow{b}= &#8211; \\overrightarrow{i} + 4\\overrightarrow{j} -\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[|\\overrightarrow{a} \u00d7 \\overrightarrow{b}| =  \\sqrt{(-1)^2 + (4)^2 + (-1)^2 }=\\sqrt{(1 + 16 +1 }=\\sqrt{18}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Area\\ of \\ parellelogram = 3\\sqrt{2} sq.units}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/HDq2BnQO4Co\" title=\"Product of Vectors (Exercise) - Part - 15\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[16.\\ If\\ \\overrightarrow{d_1}\\ =\\ 4\\overrightarrow{i}\\ +\\  2\\overrightarrow{j}\\  +\\ 3\\overrightarrow{k}\\ and\\ \\overrightarrow{d_2}\\ =\\ \\overrightarrow{i}\\ -\\  \\overrightarrow{j}\\  +\\ \\overrightarrow{k}\\ are\\ \\hspace{15cm}\\]\\[diagonals\\ of\\ a\\ parellelogram.\\ Find\\ its\\ Area\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Soln:\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{d_1}\\ =\\ 4\\overrightarrow{i}\\ +\\  2\\overrightarrow{j}\\  +\\ 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{d_1}\\ =\\ \\overrightarrow{i}\\ -\\  \\overrightarrow{j}\\  +\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{d_1}\\ \u00d7\\  \\overrightarrow{d_2}\\ =\\begin{vmatrix}\n\\overrightarrow{i} &amp;  \\overrightarrow{j} &amp; \\overrightarrow{k}\\\\\n4 &amp;  2 &amp; 3\\\\\n1 &amp; -1 &amp; 1\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( 2\\ +\\ 3) -\\overrightarrow{j}(4\\ -\\ 3)+\\overrightarrow{k}(-4\\ -\\ 2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(5) -\\overrightarrow{j}(1)\\ +\\ \\overrightarrow{k}(- 6)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{d_1}\\ \u00d7\\  \\overrightarrow{d_2}\\ =\\ 5\\overrightarrow{i}\\  -\\ \\overrightarrow{j}\\ -\\ 6\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Area\\ of \\ parellelogram =\\ \\frac{1}{2} |\\overrightarrow{d_1}\\ \u00d7\\  \\overrightarrow{d_2}|\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  \\frac{1}{2} \\sqrt{(5)^2 + (-1)^2 + (-6)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  \\frac{1}{2}\\ \\sqrt{(25\\ +\\  1\\ +\\ 36)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{2}\\ \\sqrt{62}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Area\\ of \\ parellelogram =\\ \\frac{1}{2}\\ \\sqrt{62} sq.units}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/ItJ4M5qVU84\" title=\"Product of Vectors (Exercise) - Part - 16\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[17.\\ Find\\ the\\ area\\ of\\ the\\ triangle\\ whose\\ adjacent\\ sides\\ are\\ 2\\overrightarrow{i}\\ +\\  3\\overrightarrow{j}\\ -\\ \\overrightarrow{k} and\\  \\overrightarrow{i}\\ +\\  3\\overrightarrow{j}\\ +\\ \\overrightarrow{k}.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Soln:\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}=2\\overrightarrow{i}\\ +\\  3\\overrightarrow{j}\\ -\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}=  \\overrightarrow{i}\\ +\\  3\\overrightarrow{j}\\ +\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\u00d7\\overrightarrow{b} =\\begin{vmatrix}\n\\overrightarrow{i} &amp;  \\overrightarrow{j} &amp; \\overrightarrow{k}\\\\\n2 &amp;  3 &amp; -1\\\\\n1 &amp; 3 &amp; 1\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(3\\ +\\ 3)\\ -\\ \\overrightarrow{j}(2\\ +\\ 1)\\ +\\ \\overrightarrow{k}(6\\ -\\ 3)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(6)\\  -\\ \\overrightarrow{j}(3)\\ +\\ \\overrightarrow{k}(3)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}\u00d7 \\overrightarrow{b}\\ =\\  6\\overrightarrow{i}\\ -\\ 3\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Area\\ of \\ triangle\\ =\\ \\frac{1}{2} |\\overrightarrow{a}\\ \u00d7\\  \\overrightarrow{b}|\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  \\frac{1}{2} \\sqrt{(6)^2 + (-3)^2 + (3)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  \\frac{1}{2}\\ \\sqrt{(36\\ +\\  9\\ +\\ 9)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{2}\\ \\sqrt{54}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/1fBU9hualSk\" title=\"Product of Vectors (Exercise) - Part - 17\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"aicp\"><script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block; text-align:center;\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"2812384453\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\underline{PART\\ -\\ C}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[18.\\ If\\ \\overrightarrow{a}\\ = \\ 2\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ +\\ 3\\overrightarrow{k}\\ and\\  \\overrightarrow{b}\\ =\\  \\overrightarrow{i} -\\  4\\overrightarrow{j}\\ -\\ 6\\overrightarrow{k},\\ \\hspace{15cm}\\]\\[find\\ the\\ projection\\ of\\ \\overrightarrow{a}\\ on\\ \\overrightarrow{b}\\ .\\ Also\\ find\\ the\\ angle\\ between\\ them\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Soln:\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\\ =\\  2\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ +\\ 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}\\ =\\ \\overrightarrow{i} -\\  4\\overrightarrow{j}\\ -\\ 6\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}.\\overrightarrow{b}= ( 2\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ +\\ 3\\overrightarrow{k}) .(\\overrightarrow{i} -\\  4\\overrightarrow{j}\\ -\\ 6\\overrightarrow{k})\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">= 2 ( 1 ) + 1 ( -4) + 3 ( &#8211; 6 ) <\/p>\n\n\n\n<p class=\"has-text-align-center\"> =&nbsp;&nbsp;&nbsp;&nbsp;2 &#8211; 4 &#8211; 18<\/p>\n\n\n\n<p class=\"has-text-align-center\">=&nbsp;&nbsp; &#8211; 20 <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{ \\overrightarrow{a}.\\overrightarrow{b}\\ =\\  -\\ 20}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{|a|} = \\sqrt{(2)^2 + (1)^2 + (3)^2 }=\\sqrt{4\\ +\\  1\\ +\\ 9 }=\\sqrt{14}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{|b|} = \\sqrt{(1)^2 + (-4)^2 + (-6)^2 }=\\sqrt{1\\ +\\ 16\\ +\\ 36 }=\\sqrt{53}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Projection\\  of\\ \\overrightarrow{a} on \\overrightarrow{b} = \\frac{\\overrightarrow{a}.\\overrightarrow{b}}{\\overrightarrow{|b|}} = \\frac{-20}{ \\sqrt{53}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Projection\\  of\\ \\overrightarrow{a} on \\overrightarrow{b}\\ =\\ \\frac{-20}{ \\sqrt{53}}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\  cos\\  \\theta =\\frac{\\overrightarrow{a}.\\overrightarrow{b}}{\\overrightarrow{|a|}\\overrightarrow{|b|}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  \\frac{-20}{\\sqrt{14}\\sqrt{53}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\theta = \\cos ^{-1} ( \\frac{-20}{\\sqrt{14}\\sqrt{53}})}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/xUwLVKhLj0k\" title=\"Product of Vectors (Exercise) - Part - 18\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[19.\\ Prove\\ that\\ the\\ vectors\\ \\overrightarrow{i}+2\\overrightarrow{j}+ \\overrightarrow{k},\\  \\overrightarrow{i}  + \\overrightarrow{j}- 3\\overrightarrow{k}\\ and\\ 7\\overrightarrow{i}-4\\overrightarrow{j}+\\overrightarrow{k}\\ \\hspace{15cm}\\]\\[ are\\ perpendicular\\ to\\ each\\ other.\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<p>Soln:   <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}= \\overrightarrow{i}+2\\overrightarrow{j}+ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{b}= \\overrightarrow{i}  + \\overrightarrow{j}- 3\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{c}= 7\\overrightarrow{i}-4\\overrightarrow{j}+\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}.\\overrightarrow{b}= (\\overrightarrow{i}+2\\overrightarrow{j}+ \\overrightarrow{k}) .(\\overrightarrow{i}  + \\overrightarrow{j}- 3\\overrightarrow{k})\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">   = 1(1) + 2(1) + 1(-3)<\/p>\n\n\n\n<p class=\"has-text-align-center\">=   0 <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\\ and\\ \\overrightarrow{b}\\  are\\  perpendicular\\  vectors.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{b}.\\overrightarrow{c}= (\\overrightarrow{i}  + \\overrightarrow{j}- 3\\overrightarrow{k}) .( 7\\overrightarrow{i}-4\\overrightarrow{j}+\\overrightarrow{k})\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\"> = 1(7) + 1(-4) &#8211; 3(1)<\/p>\n\n\n\n<p class=\"has-text-align-center\">= 7 &#8211; 4 &#8211; 3<\/p>\n\n\n\n<p class=\"has-text-align-center\"> =   0 <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b} and\\ \\overrightarrow{c}\\  are\\  perpendicular\\  vectors.\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">  =   7(1) -4 (2) + 1 (1)  <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{c}.\\overrightarrow{a}= (7\\overrightarrow{i}-4\\overrightarrow{j}+\\overrightarrow{k}) .(\\overrightarrow{i}+2\\overrightarrow{j}+ \\overrightarrow{k})\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\"> = 7 &#8211; 8 + 1<\/p>\n\n\n\n<p class=\"has-text-align-center\">  =   0  <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{c} and\\ \\overrightarrow{a}\\  are\\  perpendicular\\  vectors.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ The\\ three\\ vectors\\  are\\ mutually\\ perpendicular.\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/5m31gltcYek\" title=\"Product of Vectors (Exercise) - Part - 19\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[20.\\ Show\\ that\\ (\\overrightarrow{a}\\ .\\ \\overrightarrow{i})\\overrightarrow{i}\\ +\\ (\\overrightarrow{a}\\ .\\ \\overrightarrow{j})\\overrightarrow{j}\\ +\\ (\\overrightarrow{a}\\ .\\ \\overrightarrow{k})\\overrightarrow{k}\\ =\\ \\overrightarrow{a}\\ ,\\ if\\ \\overrightarrow{a}\\ is\\ any\\ vector\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Soln:\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ \\overrightarrow{a}\\ =\\ x\\overrightarrow{i}\\ +\\ y\\overrightarrow{j}\\ +\\  z\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\\ .\\ \\overrightarrow{i}\\ =\\ (x\\overrightarrow{i}\\ +\\ y\\overrightarrow{j}\\ +\\  z\\overrightarrow{k})\\ .\\ \\overrightarrow{i}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\\ .\\ \\overrightarrow{j}\\ =\\ (x\\overrightarrow{i}\\ +\\ y\\overrightarrow{j}\\ +\\  z\\overrightarrow{k})\\ .\\ \\overrightarrow{j}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ y\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\\ .\\ \\overrightarrow{k}\\ =\\ (x\\overrightarrow{i}\\ +\\ y\\overrightarrow{j}\\ +\\  z\\overrightarrow{k})\\ .\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ z\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[L.\\ H.\\ S\\ =\\ (\\overrightarrow{a}\\ .\\ \\overrightarrow{i})\\overrightarrow{i}\\ +\\ (\\overrightarrow{a}\\ .\\ \\overrightarrow{j})\\overrightarrow{j}\\ +\\ (\\overrightarrow{a}\\ .\\ \\overrightarrow{k})\\overrightarrow{k}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ x\\overrightarrow{i}\\ +\\ y\\overrightarrow{j}\\ +\\ z\\overrightarrow{k}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\overrightarrow{a}\\ =\\ R.\\ H.\\ S\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{(\\overrightarrow{a}\\ .\\ \\overrightarrow{i})\\overrightarrow{i}\\ +\\ (\\overrightarrow{a}\\ .\\ \\overrightarrow{j})\\overrightarrow{j}\\ +\\ (\\overrightarrow{a}\\ .\\ \\overrightarrow{k})\\overrightarrow{k}\\ =\\ \\overrightarrow{a}}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/RmaYfz7apZ8\" title=\"Product of Vectors (Exercise) - Part - 18\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[21\\ Find\\ the\\ area\\ of\\ the\\ triangle\\ formed\\ by\\ the\\ points\\ whose\\ position\\ vectors\\ \\hspace{15cm}\\]\\[5\\overrightarrow{i}+2\\overrightarrow{j} + 4\\overrightarrow{k}\\ ,\\  \\overrightarrow{i} +3\\overrightarrow{j}+ 2\\overrightarrow{k} and\\  -\\overrightarrow{i} &#8211; \\overrightarrow{j}+\\overrightarrow{k}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<p> Soln: <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}= 5\\overrightarrow{i}+2\\overrightarrow{j} + 4\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}= \\overrightarrow{i} +3\\overrightarrow{j}+ 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OC}=-\\overrightarrow{i} &#8211; \\overrightarrow{j}+\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB} = \\overrightarrow{OB}-\\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\overrightarrow{i} +3\\overrightarrow{j}+ 2\\overrightarrow{k}- (5\\overrightarrow{i}+2\\overrightarrow{j} + 4\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\overrightarrow{i} +3\\overrightarrow{j}+ 2\\overrightarrow{k}- 5\\overrightarrow{i}-2\\overrightarrow{j} &#8211; 4\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}=  -4\\overrightarrow{i} + \\overrightarrow{j} &#8211; 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{BC} = \\overrightarrow{OC}-\\overrightarrow{OB}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=-\\overrightarrow{i} &#8211; \\overrightarrow{j}+\\overrightarrow{k}- (\\overrightarrow{i} +3\\overrightarrow{j}+ 2\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=-\\overrightarrow{i} &#8211; \\overrightarrow{j}+\\overrightarrow{k}- \\overrightarrow{i} -3\\overrightarrow{j} &#8211; 2\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{BC}=  -2\\overrightarrow{i} &#8211; 4\\overrightarrow{j} &#8211; \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}\u00d7\\overrightarrow{BC} =\\begin{vmatrix}\n\\overrightarrow{i} &amp;  \\overrightarrow{j} &amp; \\overrightarrow{k}\\\\\n-4 &amp;  1 &amp; -2\\\\\n-2 &amp; -4 &amp; -1\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( -1 &#8211; 8) -\\overrightarrow{j}(4 &#8211; 4)+\\overrightarrow{k}(16 + 2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(-9) &#8211; 0\\overrightarrow{j}(-8)+\\overrightarrow{k}(18)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}\u00d7 \\overrightarrow{BC}= -9\\overrightarrow{i}+18\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[|\\overrightarrow{AB} \u00d7 \\overrightarrow{BC}| =  \\sqrt{(9 )^2 + (18)^2}=\\sqrt{(81 + 324 }=\\sqrt{405}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Area\\ of \\ triangle = \\frac{1}{2}|\\overrightarrow{AB} \u00d7 \\overrightarrow{BC}|\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\frac{\\sqrt{405}}{2}\\ sq. units\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/t4z25OkfNdk\" title=\"Product of Vectors (Exercise) - Part - 19\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[22.\\ \\color{red}{Find\\ the\\ area\\ of\\ the\\ triangle}\\ formed\\ by\\ the\\ points\\ whose\\ position\\ vectors\\ \\hspace{15cm}\\]\\[\\color{red}{3\\overrightarrow{i}\\ -\\ 2\\overrightarrow{j}\\ +\\ \\overrightarrow{k}\\ ,\\  \\overrightarrow{i}\\ -\\ 3\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k}\\ and\\  2\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -\\ 4\\overrightarrow{k}}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Soln:\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}\\ =\\ 3\\overrightarrow{i}\\ -\\ 2\\overrightarrow{j}\\ +\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}\\ =\\ \\overrightarrow{i}\\ -\\ 3\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OC}\\ =\\ 2\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -\\ 4\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB} = \\overrightarrow{OB}-\\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\overrightarrow{i}\\ -\\ 3\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k}\\ &#8211; (3\\overrightarrow{i}\\ -\\ 2\\overrightarrow{j}\\ +\\ \\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\overrightarrow{i}\\ -\\ 3\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k}\\ &#8211; 3\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ -\\  \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{AB}\\ =  -2\\ \\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ +\\ 4\\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{BC} = \\overrightarrow{OC}-\\overrightarrow{OB}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 2\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -\\ 4\\overrightarrow{k}\\ &#8211; (\\overrightarrow{i}\\ -\\ 3\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 2\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -\\ 4\\overrightarrow{k}\\ &#8211; \\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ -5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{BC}\\ =\\  \\overrightarrow{i}\\ +\\ 4\\overrightarrow{j}\\ -\\ 9\\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}\u00d7\\overrightarrow{BC} =\\begin{vmatrix}\n\\overrightarrow{i} &amp; \\overrightarrow{j} &amp;\\overrightarrow{k}\\\\\n-2 &amp; -1 &amp; 4\\\\\n1 &amp; 4 &amp; &#8211; 9\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(9\\ -\\ 16)\\ -\\overrightarrow{j}(18\\ -\\ 4)\\ +\\ \\overrightarrow{k}(-\\ 8 +\\ 1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(-7)\\ -\\overrightarrow{j}(14)\\  +\\ \\overrightarrow{k}(-\\ 7)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{AB}\u00d7 \\overrightarrow{BC}\\ =\\ -\\ 7\\overrightarrow{i}\\ -\\ 14\\overrightarrow{j}\\ -\\ 7\\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[|\\overrightarrow{AB} \u00d7 \\overrightarrow{BC}| =  \\sqrt{(-7 )^2\\ +\\ (-14)^2\\ +\\ ((-7)^2 }=\\sqrt{(49\\ +\\ 196\\ +\\ 49}=\\sqrt{294}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Area\\ of \\ triangle = \\frac{1}{2}|\\overrightarrow{AB} \u00d7 \\overrightarrow{BC}|\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Area\\ of \\ triangle\\ =\\frac{\\sqrt{294}}{2}\\ sq. units}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/6IJTmS8gaMo\" title=\"Product of Vectors (Exercise) - Part - 20\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[23.\\ \\color{red}{Find\\ the\\ area\\ of\\ the\\ triangle}\\ formed\\ by\\ the\\ points\\ whose\\ position\\ vectors\\ \\hspace{15cm}\\]\\[\\color{red}{2\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\ 4\\overrightarrow{k}\\ ,\\  3\\overrightarrow{i}\\ +\\ 4\\overrightarrow{j}\\ +\\ 2\\overrightarrow{k}\\ and\\  4\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k}}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}\\ =\\ 2\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\  4\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}\\ =\\ 3\\overrightarrow{i}\\ +\\ 4\\overrightarrow{j}\\  +\\ 2\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OC}\\ =\\ 4\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ +\\  3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB} = \\overrightarrow{OB}-\\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 3\\overrightarrow{i}\\ +\\ 4\\overrightarrow{j}\\  +\\ 2\\ \\overrightarrow{k}\\ &#8211; (2\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\  4\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 3\\overrightarrow{i}\\ +\\ 4\\overrightarrow{j}\\  +\\ 2\\ \\overrightarrow{k}\\ &#8211; 2\\overrightarrow{i}\\ &#8211; 3\\overrightarrow{j}\\ -\\ 4 \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{AB}\\ =\\  \\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -\\ 2\\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{BC} = \\overrightarrow{OC}-\\overrightarrow{OB}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 4\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ +\\  3\\overrightarrow{k}\\ -\\ (3\\overrightarrow{i}\\ +\\ 4\\overrightarrow{j}\\  +\\ 2\\ \\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 4\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ +\\  3\\overrightarrow{k}\\ -\\ 3\\overrightarrow{i}\\  -\\ 4\\overrightarrow{j}\\ -\\ 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{BC}\\ =\\  \\overrightarrow{i}\\ -\\ 2\\overrightarrow{j}\\ +\\ \\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}\u00d7\\overrightarrow{BC} =\\begin{vmatrix}\n\\overrightarrow{i} &amp; \\overrightarrow{j} &amp;\\overrightarrow{k}\\\\\n1 &amp; 1 &amp; -2\\\\\n1 &amp; -2 &amp; 1\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(1 -\\ 4) -\\ \\overrightarrow{j}(1\\ +\\  2)\\ +\\ \\overrightarrow{k}(-2\\ -\\ 1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(-3) -\\ \\overrightarrow{j}(3)\\ +\\ \\overrightarrow{k}(-3)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{AB}\u00d7 \\overrightarrow{BC}\\ =\\ -3\\overrightarrow{i}\\ -\\ 3\\overrightarrow{j}\\ -\\ 3\\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[|\\overrightarrow{AB} \u00d7 \\overrightarrow{BC}| =  \\sqrt{(-3 )^2 + (-3)^2 + (-3)^2 }=\\sqrt{(9\\ +\\  9\\  +\\ 9 }=\\sqrt{27}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Area\\ of \\ triangle = \\frac{1}{2}|\\overrightarrow{AB} \u00d7 \\overrightarrow{BC}|\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Area\\ of \\ triangle\\ =\\frac{\\sqrt{27}}{2}\\ sq. units}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/rVwgHHwXJGI\" title=\"Product of Vectors (Exercise) - Part - 23\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[24.\\ Find\\ the\\ unit\\ vector\\ perpendicular\\ to\\ each\\ of\\ the\\ vectors\\ \\hspace{15cm}\\]\\[3\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}+ \\overrightarrow{k}\\   and\\  2\\overrightarrow{i}\\ -\\ 5\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k}.\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Soln:\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}= 3\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}+ \\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= 2\\overrightarrow{i}\\ -\\ 5\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\u00d7\\overrightarrow{b} =\\begin{vmatrix}\n\\overrightarrow{i} &amp; \\overrightarrow{j} &amp; \\overrightarrow{k}\\\\\n3 &amp;  3 &amp;  1\\\\\n2 &amp;  &#8211; 5 &amp; 3\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( 9\\ +\\ 5)\\ -\\ \\overrightarrow{j}(9\\ -\\  2)\\ +\\ \\overrightarrow{k}(-\\ 15\\ -\\ 6)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(14)\\  -\\ \\overrightarrow{j}(7)\\ +\\ \\overrightarrow{k}(-21)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\u00d7 \\overrightarrow{b}\\ =\\ 14\\overrightarrow{i}\\ -\\ 7\\overrightarrow{j}\\ -\\ 21\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[|\\overrightarrow{a} \u00d7 \\overrightarrow{b}| =  \\sqrt{(14 )^2 + (-7)^2 + (-21)^2 }=\\sqrt{(196\\ +\\ 49\\ +\\  441}\\ =\\ \\sqrt{686}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ n^\\wedge  =\\frac{\\overrightarrow{a}\u00d7 \\overrightarrow{b}}{|\\overrightarrow{a} \u00d7 \\overrightarrow{b}|}\\ =\\  \\frac{14\\overrightarrow{i} -\\ 7\\overrightarrow{j}\\ -\\ 21\\overrightarrow{k}}{\\sqrt{686}} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{n^\\wedge  =  \\frac{14\\overrightarrow{i} -\\ 7\\overrightarrow{j}\\ -\\ 21\\overrightarrow{k}}{\\sqrt{686}}}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/05PnUPZhdlE\" title=\"Product of Vectors (Exercise) - Part - 21\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[25.\\ Find\\ the\\ unit\\ vector\\ perpendicular\\ to\\ each\\ of\\ the\\ vectors\\   \\overrightarrow{i} &#8211; \\overrightarrow{j}+ 3\\overrightarrow{k}   and\\  2\\overrightarrow{i}+ 3\\overrightarrow{j} -\\overrightarrow{k}.\\ \\hspace{10cm}\\]\\[Also\\ find\\ the\\ sine\\ of\\ the\\ angle\\ between\\ the\\ vectors .\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<p>Soln: <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}= \\overrightarrow{i} &#8211; \\overrightarrow{j}+ 3\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= 2\\overrightarrow{i}+ 3\\overrightarrow{j} -\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\u00d7\\overrightarrow{b} =\\begin{vmatrix}\n\\overrightarrow{i} &amp; \\overrightarrow{j} &amp; \\overrightarrow{k}\\\\\n1 &amp;  -1 &amp;  3\\\\\n2 &amp;  3 &amp; -1\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( 1 &#8211; 9) -\\overrightarrow{j}(-1 &#8211; 6)+\\overrightarrow{k}(3 + 2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(-8) -\\overrightarrow{j}(-7)+\\overrightarrow{k}(5)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\u00d7 \\overrightarrow{b}= -8\\overrightarrow{i} + 7\\overrightarrow{j}+5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[|\\overrightarrow{a} \u00d7 \\overrightarrow{b}| =  \\sqrt{(-8 )^2 + (-7)^2 + (5)^2 }=\\sqrt{(64 + 49 + 25}=\\sqrt{138}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ n^\\wedge  =\\frac{\\overrightarrow{a}\u00d7 \\overrightarrow{b}}{|\\overrightarrow{a} \u00d7 \\overrightarrow{b}|}= \\frac{-8\\overrightarrow{i} + 7\\overrightarrow{j}+5\\overrightarrow{k}}{\\sqrt{138}} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ n^\\wedge  = \\frac{-8\\overrightarrow{i} + 7\\overrightarrow{j}+5\\overrightarrow{k}}{\\sqrt{138}} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{|a|} = \\sqrt{(1)^2 + (-1)^2 + (3)^2 }=\\sqrt{(1 + 1 + 9 }=\\sqrt{11}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{|b|} = \\sqrt{(2)^2 + (3)^2 + (-1)^2 }=\\sqrt{(4 + 9 + 1 }=\\sqrt{14}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\  sin\\  \\theta =\\frac{|\\overrightarrow{a} \u00d7 \\overrightarrow{b}|}{\\overrightarrow{|a|}\\overrightarrow{|b|}}= \\frac{\\sqrt{138}}{\\sqrt{11}\\sqrt{14}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\  sin\\  \\theta =\\frac{\\sqrt{138}}{\\sqrt{11}\\sqrt{14}}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/t7p8WM8XLiY\" title=\"Product of Vectors (Exercise) - Part - 22\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[26.\\ The\\  forces\\ 2\\overrightarrow{i}\\ -\\ 5\\overrightarrow{j}\\ + \\ 6\\overrightarrow{k}\\ ,\\  -\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ &#8211; \\ \\overrightarrow{k}\\ and\\ 2\\overrightarrow{i}\\ +\\ 7\\overrightarrow{j}\\ act\\ on\\ a\\  particle\\ \\hspace{15cm}\\]\\[and\\ displace\\ it\\ from\\ the\\ point\\ 4\\overrightarrow{i}\\ -\\ 3\\overrightarrow{j}\\ -\\ 2\\overrightarrow{k}\\ to\\ the\\ point\\ 6\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ &#8211; \\ 3\\overrightarrow{k}.\\ Find\\ the\\ total\\ work\\ done\\ by\\ the\\ forces\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Soln:\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F_1}= 2\\overrightarrow{i}\\ -\\ 5\\overrightarrow{j}\\ + \\ 6\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F_2}= -\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ &#8211; \\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F_3}= 2\\overrightarrow{i}\\ +\\ 7\\overrightarrow{j} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F}=\\overrightarrow{F_1} + \\overrightarrow{F_2}\\ +\\ \\overrightarrow{F_3}\\ =\\ 3\\overrightarrow{i}+ 2\\overrightarrow{j}- 3\\overrightarrow{k} + \\overrightarrow{i}+ 7\\overrightarrow{j}+7\\overrightarrow{k}\\ +\\ 2\\overrightarrow{i}\\ +\\ 7\\overrightarrow{j}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{F}= 3\\overrightarrow{i}\\ + 4\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}= 4\\overrightarrow{i}\\ -\\ 3\\overrightarrow{j}\\ -\\ 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}= 6\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ &#8211; \\ 3\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow {d}= \\overrightarrow {OB}- \\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=6\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ &#8211; \\ 3\\overrightarrow{k}- (4\\overrightarrow{i}\\ -\\ 3\\overrightarrow{j}\\ -\\ 2\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=6\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ &#8211; \\ 3\\overrightarrow{k}\\ &#8211; 4\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\  2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{d}\\ =\\ 2\\overrightarrow{i}\\ +\\ 4\\overrightarrow{j}\\ -\\ \\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Work\\  done =  \\overrightarrow{F}.\\overrightarrow{d}= (3\\overrightarrow{i}\\ + 4\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k}) .(2\\overrightarrow{i}\\ +\\ 4\\overrightarrow{j}\\ -\\ \\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\   3 ( 2 )\\ +\\ 4 ( 4 )\\ + 5 ( -1 )\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 6\\ +\\  16\\ -\\ 5\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Work\\ done\\ =\\ 17\\ units}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/A37fPmFH3-o\" title=\"Product of vectors (Exercise Problems) - Part - 23\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[27.\\ The\\  forces\\ 3\\overrightarrow{i}\\ +\\ 5\\overrightarrow{j}\\ &#8211; \\ 2\\overrightarrow{k}\\ and\\  2\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ &#8211; \\ 5\\overrightarrow{k}\\ displaces\\ a\\ particle\\ \\hspace{15cm}\\]\\[from\\ the\\ point\\ (1, 2, -1)\\ to\\ the\\ point\\ (5, -3, 4).\\ Find\\ the\\ total\\ work\\ done\\ by\\ the\\ force.\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Soln:\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F_1}= 3\\overrightarrow{i}\\ +\\ 5\\overrightarrow{j}\\ &#8211; \\ 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F_2}= 2\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ &#8211; \\ 5\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F}=\\overrightarrow{F_1} + \\overrightarrow{F_2} = 3\\overrightarrow{i}\\ +\\ 5\\overrightarrow{j}\\ &#8211; \\ 2\\overrightarrow{k} + 2\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ &#8211; \\ 5\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{F}\\ =\\ 5\\overrightarrow{i}\\ + 8\\overrightarrow{j}\\ -\\ 7\\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}= \\overrightarrow{i}+ 2\\overrightarrow{j}\\ -\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}\\ =\\ 5\\overrightarrow{i}\\ -3\\overrightarrow{j}\\ +\\ 4\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow {d}= \\overrightarrow {OB}- \\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 5\\overrightarrow{i}\\ -3\\overrightarrow{j}\\ +\\ 4\\overrightarrow{k}\\ &#8211; (\\overrightarrow{i}+ 2\\overrightarrow{j}\\ -\\ \\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=5\\overrightarrow{i}\\ -3\\overrightarrow{j}\\ +\\ 4\\overrightarrow{k}\\ &#8211; \\overrightarrow{i}\\ &#8211; 2\\overrightarrow{j}\\ + \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{d}\\ =\\ 4\\overrightarrow{i}\\ &#8211; 5\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Work\\  done =  \\overrightarrow{F}.\\overrightarrow{d}\\ = (5\\overrightarrow{i}\\ + 8\\overrightarrow{j}\\ -\\ 7\\overrightarrow{k}) .(4\\overrightarrow{i}\\ &#8211; 5 \\overrightarrow{j}\\ +\\ 5\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\   5( 4 )\\ +\\ 8 ( -5 )\\ -7 ( 5 )\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 20\\ -\\  40\\ -\\ 35\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Work\\ done\\ =\\ -\\ 55\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Work\\ done\\ =\\ 55\\ units}\\ (by\\ taking\\ positive\\ value)\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/MlgUndVKLLk\" title=\"Product of Vectors (Exercise) - Part - 24\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[28.\\ \\color{red}{Find\\ the\\ torque\\ of\\ the\\ force\\ 3\\overrightarrow{i}+4\\overrightarrow{j}+5\\overrightarrow{k}\\ acting\\ through\\ the\\ point}\\      \\hspace{10cm}\\]\\[\\color{red}{\\overrightarrow{i}-2\\overrightarrow{j}+3\\overrightarrow{k} about\\ the\\ point\\  4\\overrightarrow{i}- 3\\overrightarrow{j}+\\overrightarrow{k}.}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Soln:\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F}= 3\\overrightarrow{i} +4\\overrightarrow{j} + 5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OP}= \\overrightarrow{i} &#8211; 2\\overrightarrow{j} + 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}= 4\\overrightarrow{i} -3\\overrightarrow{j} +\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{r}= \\overrightarrow{AP} = \\overrightarrow{OP}-\\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\overrightarrow{i} &#8211; 2\\overrightarrow{j} + 3\\overrightarrow{k}- (4\\overrightarrow{i} -3\\overrightarrow{j} +\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\overrightarrow{i} &#8211; 2\\overrightarrow{j} + 3\\overrightarrow{k}- 4\\overrightarrow{i} +3\\overrightarrow{j} -\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{r}=  -3\\overrightarrow{i} + \\overrightarrow{j} + 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Moment = \\overrightarrow{r}\u00d7 \\overrightarrow{F}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\begin{vmatrix}\n\\overrightarrow{i} &amp;  \\overrightarrow{j} &amp; \\overrightarrow{k}\\\\\n-3 &amp; 1 &amp; 2\\\\\n3 &amp; 4 &amp; 5\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( 5 &#8211; 8) -\\overrightarrow{j}(-15 &#8211; 6)+\\overrightarrow{k}(-12 &#8211; 3)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{r}\u00d7 \\overrightarrow{F}= -3\\overrightarrow{i}+ 21\\overrightarrow{j}-15\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Magnitude\\ of \\ Moment = |\\overrightarrow{r} \u00d7 \\overrightarrow{F}|\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  \\sqrt{(-3)^2 + (21)^2 + (-15)^2 }=\\sqrt{(9 + 441+ 225 }=\\sqrt{675}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Magnitude\\ of \\ Moment = \\sqrt{675}}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/nTwRHUzi1P8\" title=\"Product of Vectors (Exercise) - 25\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"aicp\">\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"autorelaxed\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"3040040190\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>= 1(1) + 1(1) + 0 = 1 + 1 = 2 2(p) + 1(3) &#8211; 5 (-2) = 0 2p + 3 +10 = 0 2p + 13 = 0 Soln: = 2 ( 1 ) + 1 ( -4) + 3 ( &#8211; 6 ) =&nbsp;&nbsp;&nbsp;&nbsp;2 &#8211; 4 &#8211; 18 =&nbsp;&nbsp; &#8211; 20 [&hellip;]<\/p>\n","protected":false},"author":187055548,"featured_media":20305,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center 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SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>N \u2013 2.2 \u2013 Product of two vectors \u2013 Exercise Problems with solutions - YANAMTAKSHASHILA<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/yanamtakshashila.com\/?p=15051\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"N \u2013 2.2 \u2013 Product of two vectors \u2013 Exercise Problems with solutions - YANAMTAKSHASHILA\" \/>\n<meta property=\"og:description\" content=\"= 1(1) + 1(1) + 0 = 1 + 1 = 2 2(p) + 1(3) &#8211; 5 (-2) = 0 2p + 3 +10 = 0 2p + 13 = 0 Soln: = 2 ( 1 ) + 1 ( -4) + 3 ( &#8211; 6 ) =&nbsp;&nbsp;&nbsp;&nbsp;2 &#8211; 4 &#8211; 18 =&nbsp;&nbsp; &#8211; 20 [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/yanamtakshashila.com\/?p=15051\" \/>\n<meta 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