{"id":12526,"date":"2021-03-01T16:47:06","date_gmt":"2021-03-01T11:17:06","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=12526"},"modified":"2024-04-20T11:33:33","modified_gmt":"2024-04-20T06:03:33","slug":"2-2-scalar-product-of-vectors","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=12526","title":{"rendered":"2.2 PRODUCT OF VECTORS"},"content":{"rendered":"\n<h4 class=\"wp-block-heading has-vivid-red-color has-text-color\">SCALAR PRODUCT<\/h4>\n\n\n\n<figure class=\"wp-block-image aligncenter\"><img data-recalc-dims=\"1\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/files.askiitians.com\/cdn1\/images\/2014108-144310695-8606-dot-product-image.png?ssl=1\" alt=\"Multiplication of vectors -Study Material for IIT JEE | askIITians\"\/><\/figure>\n\n\n\n<h5 class=\"wp-block-heading has-vivid-cyan-blue-color has-text-color\"><strong>Definition:&nbsp;<\/strong>  <\/h5>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let \\overrightarrow{a}\\ and  \\overrightarrow{b}\\  be\\ two\\ non\\ zero\\ vectors\\ inclined\\ at\\  an\\  angle\\ \\theta.\\] \\[Then\\ the\\ scalar\\ product\\ of\\  \\overrightarrow{a} and \\overrightarrow{b} is\\  denoted\\  by\\]\\[\\overrightarrow{a}.\\overrightarrow{b} and\\ is\\ defined\\ as\\overrightarrow{a}.\\overrightarrow{b}=\\overrightarrow{|a|}\\overrightarrow{|b|}\\cos\\theta \\]<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=\"aicp\" --=\"\">\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<!-- sidebar ad 1 -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:326px;height:280px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"6703350399\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n<\/div>\n\n\n\n<h5 class=\"wp-block-heading has-vivid-cyan-blue-color has-text-color\"><strong>Properties of Scalar Product:<\/strong><\/h5>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.    \\overrightarrow{a}\\ and\\overrightarrow{b}are\\ perpendicular\\ vectors\\  if\\ and\\ only\\ if  \\overrightarrow{a}.\\overrightarrow{b}=0. \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.    \\overrightarrow{i},\\overrightarrow{j}\\  and\\  \\overrightarrow{k}\\ are\\  the\\ unit\\ vectors\\ along\\ the\\ x, y, z\\ axis\\  respectively   \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Then\\ i)  \\overrightarrow{i}.\\overrightarrow{i}= \\overrightarrow{j}.\\overrightarrow{j}=\\overrightarrow{k}.\\overrightarrow{k}=1  \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ ii)  \\overrightarrow{i}.\\overrightarrow{j}= \\overrightarrow{j}.\\overrightarrow{k}=\\overrightarrow{k}.\\overrightarrow{i}= \\overrightarrow{j}.\\overrightarrow{i}= \\overrightarrow{k}.\\overrightarrow{j}=\\overrightarrow{i}.\\overrightarrow{k}=0  \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3. \\  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\">\\[4. \\  cos\\  \\theta =\\frac{\\overrightarrow{a}.\\overrightarrow{b}}{\\overrightarrow{|a|}\\overrightarrow{|b|}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\  \\overrightarrow{a}= a_1\\overrightarrow{i}\\ + a_2\\overrightarrow{j}+ a_3\\overrightarrow{k}\\ and\\ \\overrightarrow{b}= b_1\\overrightarrow{i}\\ + b_2\\overrightarrow{j}+ b_3\\overrightarrow{k}\\ . Then\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}.\\overrightarrow{b}= a_1b_1 + a_2b_2 + a_3b_3\\  (using\\  property\\  2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 1\\ .}\\ \\color {red} {Find\\  the\\  scalar\\  product\\  of}\\  2\\overrightarrow{i} -4\\overrightarrow{j}\\ + 8\\overrightarrow{k}\\  and\\  \\overrightarrow{i}\\  +6\\overrightarrow{j}+ 12\\overrightarrow{k}\\ \\hspace{10cm}\\]<\/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} -4\\overrightarrow{j}\\ + 8\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= \\overrightarrow{i}\\ +\\ 6\\overrightarrow{j}+ 12\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}.\\overrightarrow{b}= (2\\overrightarrow{i} -4\\overrightarrow{j}\\ + 8\\overrightarrow{k}) .( \\overrightarrow{i} +6\\overrightarrow{j}+ 12\\overrightarrow{k})\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">=  2(1)  &#8211; 4 (6)  +  8 (12)<\/p>\n\n\n\n<p class=\"has-text-align-center\"> =   2 &#8211; 24 + 96<\/p>\n\n\n\n<p class=\"has-text-align-center\"> =   74<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\boxed{\\overrightarrow{a}.\\overrightarrow{b}= 74}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/D2TkpS76VsA\" title=\"Product of Vectors - Part - 1\" 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} {Example\\ 2\\ .}\\ \\color {red} {Find\\  the\\  Dot\\  product\\  of}\\  \\overrightarrow{i}+\\overrightarrow{j}\\ +\\ \\overrightarrow{k}\\  ,\\  \\overrightarrow{i}\\ + 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}\\ +\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= \\overrightarrow{i}\\ + 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}.\\overrightarrow{b}= (\\overrightarrow{i}+\\overrightarrow{j}\\ +\\ \\overrightarrow{k}) .(\\overrightarrow{i}\\ + 3\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 1(1)\\ +\\ 1(0)\\ +\\ 1(3)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 1\\ +\\ 0\\ +\\ 3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 4\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\boxed{\\overrightarrow{a}.\\overrightarrow{b}\\ =\\ 4}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/scvG78HhlAc\" title=\"Product of Vectors - 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\"><\/div>\n\n\n\n<div class=\"aicp\" --=\"\"><script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/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<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 3\\ .}\\ \\color {red} {Write\\ the\\ condition\\ for\\ two\\ vectors\\ to\\ be\\ perpendicular}\\ \\hspace{15cm}\\]<\/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{b}\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 4\\ .}\\ \\color {red} {Show\\ that\\ the\\ vectors}\\ 2\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ &#8211; 2\\overrightarrow{k} and\\  3\\overrightarrow{i}\\ + 2\\overrightarrow{j}\\ +\\ 6\\overrightarrow{k} are\\ perpendicular\\ to\\ each\\ other \\]<\/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}+ 3\\overrightarrow{j}- 2\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= 3\\overrightarrow{i}+2\\overrightarrow{j}+ 6\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}.\\overrightarrow{b}= (2\\overrightarrow{i}+ 3\\overrightarrow{j}- 2\\overrightarrow{k}) .(3\\overrightarrow{i}+2\\overrightarrow{j}+ 6\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 2(3)\\ +\\ 3(2)\\ -\\ 2(6)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 6\\ +\\ 6\\ -\\ 12\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/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=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/YFIFFGAjxCw\" title=\"Product of Vectors - 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=\"aicp\" --=\"\"><script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"fluid\" data-ad-layout-key=\"-6t+ed+2i-1n-4w\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"9770958327\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 5\\ .}\\ \\color {red} {Show\\ that\\ the\\ vectors}\\ 4\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -\\  3\\overrightarrow{k}\\ and\\  3\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k} are\\ perpendicular\\]<\/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}\\ =\\ 4\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ &#8211; 3\\ \\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= 3\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}.\\overrightarrow{b}\\ =\\ (4\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -\\ 3\\overrightarrow{k}) .(3\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 4(3)\\ +\\ 1(3)\\ -\\ 3(5)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 12\\ +\\ 3\\ -\\ 15\\]<\/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\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 6\\ .}\\ \\color {red} {Find\\ the\\ value\\ of\\ m}\\ if\\ the\\ vectors\\ 3\\overrightarrow{i} -\\overrightarrow{j} + 5\\overrightarrow{k} and\\  -6\\overrightarrow{i}+ m\\overrightarrow{j}+4\\overrightarrow{k} are\\ perpendicular\\ \\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{a}= 3\\overrightarrow{i} -\\overrightarrow{j} + 5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{b}= -6\\overrightarrow{i}+ m\\overrightarrow{j}+4\\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\">\\[(3\\overrightarrow{i} -\\overrightarrow{j} + 5\\overrightarrow{k}).(-6\\overrightarrow{i}+ m\\overrightarrow{j}+4\\overrightarrow{k}) = 0 .\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">           3(-6) &#8211; 1(m) + 5 (4)  =   0<\/p>\n\n\n\n<p class=\"has-text-align-center\">-18 &#8211; m  + 20  = 0<\/p>\n\n\n\n<p class=\"has-text-align-center\">-m + 2   =  0<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{m\\ =\\  2}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/UjKeDiVbwRQ\" title=\"Product of Vectors - 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} {Example\\ 7\\ .}\\ \\color {red} {\\ Prove\\ that}\\ the\\ vectors\\ \\overrightarrow{i}-\\overrightarrow{j}+ 2\\overrightarrow{k},\\   \\overrightarrow4{j}+ 2\\overrightarrow{k}and\\ \n  -10\\overrightarrow{i}-2\\overrightarrow{j}+4\\overrightarrow{k} are\\ mutually\\ perpendicular.\\ \\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{a}= \\overrightarrow{i}- \\overrightarrow{j}+ 2\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{b}= + 4\\overrightarrow{j}+2\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{c}= -10\\overrightarrow{i}- 2\\overrightarrow{j}+ 4\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}.\\overrightarrow{b}= (\\overrightarrow{i}- \\overrightarrow{j}+ 2\\overrightarrow{k}) .(4\\overrightarrow{j}+ 2\\overrightarrow{k})\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\"> = 1(0) &#8211; 1(4) + 2 (2)<\/p>\n\n\n\n<p class=\"has-text-align-center\">=   0<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{a}\\ and\\ \\overrightarrow{b}\\  are\\  perpendicular\\  vectors}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{b}.\\overrightarrow{c}= (0\\overrightarrow{i}+ 4\\overrightarrow{j}+ 2\\overrightarrow{k}) .(-10\\overrightarrow{i}-2\\overrightarrow{j}+4\\overrightarrow{k})\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">  = 0(-10) + 4(-2) + 2 <\/p>\n\n\n\n<p class=\"has-text-align-center\">= 0 &#8211; 8 + 8 <\/p>\n\n\n\n<p class=\"has-text-align-center\">=   0<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{b} and\\ \\overrightarrow{c}\\  are\\  perpendicular\\  vectors}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{c}.\\overrightarrow{a}= (-10\\overrightarrow{i}- 2\\overrightarrow{j}+ 4\\overrightarrow{k}) .(\\overrightarrow{i}-\\overrightarrow{j}+ 2\\overrightarrow{k})\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">=  &#8211; 10(1) -2 (-1) + 4 (2)<\/p>\n\n\n\n<p class=\"has-text-align-center\">= &#8211; 10 + 2 + 8<\/p>\n\n\n\n<p class=\"has-text-align-center\"> = 0<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{c} and\\ \\overrightarrow{a}\\  are\\  perpendicular\\  vectors}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ The\\ three\\ vectors\\  are\\ mutually\\ perpendicular.\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/2O7lmZDTavg\" title=\"Product of Vectors - Part - 6\" 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<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"autorelaxed\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"4869133702\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\underline{Projection\\ of\\  \\overrightarrow{a}\\  on\\  \\overrightarrow{b}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 8\\ .}\\ \\color {red} {Find\\ the\\ projection\\ of\\ the\\ vector}\\ 3\\overrightarrow{i}+ 4\\overrightarrow{j}- 5\\overrightarrow{k} on\\  the\\ vector\\  \\overrightarrow{i}+ 2\\overrightarrow{j}+2\\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{a}= 3\\overrightarrow{i}+ 4\\overrightarrow{j}- 5\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= \\overrightarrow{i}+2 \\overrightarrow{j}+ 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{(3\\overrightarrow{i}+ 4\\overrightarrow{j}- 5\\overrightarrow{k}).(\\overrightarrow{i}+2\\overrightarrow{j}+ 2\\overrightarrow{k})}{\\sqrt{(1)^2 + (2)^2 + (2)^2 }}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  \\frac{3(1)+ 4(2)- 5(2)}{\\sqrt{(1 + 4 + 4 }}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  \\frac{3+ 8- 10}{\\sqrt{9}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Projection\\  of\\ \\overrightarrow{a} on \\overrightarrow{b} =\\frac{1}{3}}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/eIW6_5_mu-8\" title=\"Product of Vectors - Part - 7\" 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} {Example\\ 9\\ .}\\ \\color {red} {Find\\ the\\ projection\\ of\\ the\\ vector}\\ 3\\overrightarrow{i}+ 4\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k} on\\  the\\ vector\\  \\overrightarrow{i}+ 2\\overrightarrow{j}\\ +\\ 6\\overrightarrow{k} \\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{18cm}\\ April\\ 2024\\]<\/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}= 3\\overrightarrow{i}+ 4\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= \\overrightarrow{i}\\ +\\ 2 \\overrightarrow{j}\\ + 6\\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{(3\\overrightarrow{i}\\ +\\ 4\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k}).(\\overrightarrow{i}+2\\overrightarrow{j}\\ +\\ 6\\overrightarrow{k})}{\\sqrt{(1)^2\\ +\\ (2)^2\\ +\\ (6)^2 }}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  \\frac{3(1)\\ +\\ 4(2)\\ +\\ 5(6)}{\\sqrt{(1 + 4 + 36 }}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  \\frac{3\\ +\\ 8\\ +\\ 30}{\\sqrt{41}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Projection\\  of\\ \\overrightarrow{a} on \\overrightarrow{b} =\\frac{41}{\\sqrt{41}}}\\]<\/div>\n\n\n\n<h2 class=\"wp-block-heading has-vivid-purple-color has-text-color has-medium-font-size\">Angle between two vectors using scalar product<\/h2>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\underline{Angle\\  between\\  two\\  vectors\\  \\overrightarrow{a} and\\ \\overrightarrow{b}}: \\  cos\\  \\theta =\\frac{\\overrightarrow{a}.\\overrightarrow{b}}{\\overrightarrow{|a|}\\overrightarrow{|b|}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 10\\ .}\\ \\color {red} {Find\\ the\\ angle\\ between\\ vectors}\\ 3\\overrightarrow{i}+ 4\\overrightarrow{j}- 2\\overrightarrow{k} and\\  2\\overrightarrow{i} -3\\overrightarrow{j}- 5\\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{a}= 3\\overrightarrow{i}+ 4\\overrightarrow{j}- 2\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= 2\\overrightarrow{i}-3 \\overrightarrow{j}- 5\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}.\\overrightarrow{b}= (3\\overrightarrow{i}+ 4\\overrightarrow{j}- 2\\overrightarrow{k}) .(2\\overrightarrow{i}-3 \\overrightarrow{j}- 5\\overrightarrow{k})\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">                     = &nbsp; 3 ( 2 ) + 4 ( &#8211; 3 ) &#8211; 2 ( &#8211; 5 )<\/p>\n\n\n\n<p class=\"has-text-align-center\">=&nbsp;&nbsp;&nbsp; 6&nbsp; &#8211; 12 + 10<\/p>\n\n\n\n<p class=\"has-text-align-center\">= 4<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{ \\overrightarrow{a}.\\overrightarrow{b}= 4}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{|a|} = \\sqrt{(3)^2 + (4)^2 + (-2)^2 }=\\sqrt{(9 + 16 +4 }=\\sqrt{29}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{|b|} = \\sqrt{(2)^2 + (-5)^2 + (-3)^2 }=\\sqrt{(4 + 25 +9 }=\\sqrt{38}\\]<\/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{4}{\\sqrt{29}\\sqrt{38}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\theta = \\cos ^-1 ( \\frac{4}{\\sqrt{29}\\sqrt{38}})}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/7tSi-KtSEZo\" title=\"Product of Vectors - Part - 8\" 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} {Example\\ 11\\ .}\\ \\color {red} {Find\\ the\\ projection\\ of\\ the\\ vector}\\ 3\\overrightarrow{i}+ \\overrightarrow{j}- 2\\overrightarrow{k} on\\  7\\overrightarrow{i}+ \\overrightarrow{j}+2\\overrightarrow{k}\\ .\\hspace{8cm}\\]\\[ Also\\ \\color {red} {find\\ the\\ angle\\ between\\ them}\\ \\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{a}= 3\\overrightarrow{i}+ \\overrightarrow{j}- 2\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= 7\\overrightarrow{i}+\\overrightarrow{j}+ 2\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}.\\overrightarrow{b}= (3\\overrightarrow{i}+ \\overrightarrow{j}- 2\\overrightarrow{k}) .(7\\overrightarrow{i}+\\overrightarrow{j}+ 2\\overrightarrow{k})\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">=3 ( 7 ) + 1 ( 1) &#8211; 2 ( 2 )&nbsp;<\/p>\n\n\n\n<p class=\"has-text-align-center\">&nbsp;&nbsp; =&nbsp;&nbsp;&nbsp;&nbsp;21&nbsp;+ 1 &#8211; 4<\/p>\n\n\n\n<p class=\"has-text-align-center\">=&nbsp;&nbsp; 18.<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{ \\overrightarrow{a}.\\overrightarrow{b}= 18}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{|a|} = \\sqrt{(3)^2 + (1)^2 + (-2)^2 }=\\sqrt{(9 + 1 +4 }=\\sqrt{14}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{|b|} = \\sqrt{(7)^2 + (1)^2 + (2)^2 }=\\sqrt{(49 + 1 + 4 }=\\sqrt{54}\\]<\/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{18}{ \\sqrt{54}}\\]<\/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{18}{\\sqrt{14}\\sqrt{54}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\theta = \\cos ^-1 ( \\frac{18}{\\sqrt{14}\\sqrt{54}})}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/kYs6ieaAEsI\" title=\"Product of Vectors - 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<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"autorelaxed\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"4869133702\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-red-color has-text-color\">APPLICATION OF SCALAR PRODUCT<\/h3>\n\n\n\n<h4 class=\"wp-block-heading has-vivid-purple-color has-text-color\">Work done<\/h4>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img data-recalc-dims=\"1\" fetchpriority=\"high\" decoding=\"async\" width=\"334\" height=\"151\" data-attachment-id=\"30911\" data-permalink=\"https:\/\/yanamtakshashila.com\/?attachment_id=30911\" data-orig-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/06\/image.png?fit=334%2C151&amp;ssl=1\" data-orig-size=\"334,151\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"image\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/06\/image.png?fit=334%2C151&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/06\/image.png?resize=334%2C151&#038;ssl=1\" alt=\"\" class=\"wp-image-30911\"\/><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Work\\ done = \\overrightarrow{F}.\\overrightarrow{d}\\ where\\ \\overrightarrow{d}= \\overrightarrow {OB}- \\overrightarrow{OA}\\]<\/div>\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\">\\[\\color {purple} {Example\\ 12\\ .}\\ \\color {red} {Find\\ the\\ work\\ done}\\ by\\ the\\ force\\  3\\overrightarrow{i}+ 5\\overrightarrow{j}+ 7\\overrightarrow{k},\\ \\hspace{8cm}\\]\\[when\\ the\\ displacement\\ is\\  2\\overrightarrow{i}- \\overrightarrow{j} +\\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}= 3\\overrightarrow{i}+ 5\\overrightarrow{j}+7\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{d}= 2\\overrightarrow{i}- \\overrightarrow{j}+\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Work\\  done =  \\overrightarrow{F}.\\overrightarrow{d}= (3\\overrightarrow{i}+ 5\\overrightarrow{j}+ 7\\overrightarrow{k}) .(2\\overrightarrow{i}- \\overrightarrow{j}+ \\overrightarrow{k})\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">= 3 ( 2 ) + 5 ( &#8211; 1 ) + 7 ( 1 )<\/p>\n\n\n\n<p class=\"has-text-align-center\">=&nbsp;&nbsp;&nbsp; 6&nbsp; &#8211; 5&nbsp; +&nbsp; 7<\/p>\n\n\n\n<p class=\"has-text-align-center\">Work done = 8 units<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Work\\ done\\  =\\  8\\ units}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/v5aT8tnB_Fc\" title=\"Product of Vectors - Part - 10\" 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} {Example\\ 13\\ .}\\ A\\ particle\\ acted\\ on\\  by\\ the\\ forces\\ 3\\overrightarrow{i}+ 2\\overrightarrow{j}- 3\\overrightarrow{k} and\\  \\overrightarrow{i}+ 7\\overrightarrow{j}+7\\overrightarrow{k}\\ \\hspace{5cm}\\]\\[ acting\\ on\\ the\\ particle\\ displaces\\ the\\ particle\\ from\\ the\\  point\\  \\overrightarrow{i}+ 2\\overrightarrow{j}+ 3\\overrightarrow{k} to\\  the\\ point\\  3\\overrightarrow{i}- 5\\overrightarrow{j}+4\\overrightarrow{k}\\ .\\ \\hspace{5cm}\\]\\[\\color {red} {Find\\ the\\ total\\ work\\ done\\ by\\ the\\ forces}.\\ \\hspace {10cm}\\]<\/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{F_1}= 3\\overrightarrow{i}+ 2\\overrightarrow{j}- 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F_2}= \\overrightarrow{i}+ 7\\overrightarrow{j}+7\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F}=\\overrightarrow{F_1} + \\overrightarrow{F_2} = 3\\overrightarrow{i}+ 2\\overrightarrow{j}- 3\\overrightarrow{k} + \\overrightarrow{i}+ 7\\overrightarrow{j}+7\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{F}= 4\\overrightarrow{i}+ 9\\overrightarrow{j}+4\\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}= \\overrightarrow{i}+ 2\\overrightarrow{j}+ 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}= 3\\overrightarrow{i}-5\\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\">\\[=3\\overrightarrow{i}\\ &#8211; 5\\overrightarrow{j} + 4\\overrightarrow{k}- (\\overrightarrow{i}\\ + 2\\overrightarrow{j}+3\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\overrightarrow{i}\\ &#8211; \\overrightarrow{j} + 3\\overrightarrow{k}- \\overrightarrow{i}\\ &#8211; 2\\overrightarrow{j}- 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{d}= 2\\overrightarrow{i}- 7\\overrightarrow{j}+\\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Work\\  done =  \\overrightarrow{F}.\\overrightarrow{d}= (4\\overrightarrow{i}+ 9\\overrightarrow{j}+ 4\\overrightarrow{k}) .(2\\overrightarrow{i}-7 \\overrightarrow{j}+ 1\\overrightarrow{k})\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">= 4 ( 2 ) + 9 ( &#8211; 7 ) + 4 ( 1 )<\/p>\n\n\n\n<p class=\"has-text-align-center\">=&nbsp;&nbsp;&nbsp; 8&nbsp; &#8211; 63&nbsp; +&nbsp; 4<\/p>\n\n\n\n<p class=\"has-text-align-center\">Work done&nbsp; =&nbsp; &#8211;51 units<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Work\\ done\\ =\\ 51\\ units}\\ (by\\ taking\\ positive\\ value)\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/jYRS4wFU1ak\" title=\"Product of Vectors - Part - 11\" 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<!-- wide skyscraper -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:300px;height:600px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"9987820756\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 14\\ .}A\\ particle\\ acted\\ on\\  by\\ the\\ forces\\ 4\\overrightarrow{i}+ 3\\overrightarrow{j}+ \\overrightarrow{k} and\\  2\\overrightarrow{i}+ 7\\overrightarrow{j}-2\\overrightarrow{k}\\ \\hspace{5cm}\\]\\[is\\ displaced\\ from\\ the\\ point\\ ( 1, 1,  1 )\\  to\\ the\\  point\\  ( 2, &#8211; 3, 5 ).\\]\\[\\color {red} {Find\\ the\\ total\\ work\\ done}\\ \\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{F_1}= 4\\overrightarrow{i}+ 3\\overrightarrow{j}+ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F_2}= 2\\overrightarrow{i}+ 7\\overrightarrow{j}-2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F}=\\overrightarrow{F_1} + \\overrightarrow{F_2} = 4\\overrightarrow{i}+ 3\\overrightarrow{j}+ \\overrightarrow{k} + 2\\overrightarrow{i}+ 7\\overrightarrow{j}-2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{F}= 6\\overrightarrow{i}+ 10\\overrightarrow{j}-\\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}= \\overrightarrow{i}+ \\overrightarrow{j}+ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}= 2\\overrightarrow{i}-3\\overrightarrow{j}+ 5\\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\">\\[=2\\overrightarrow{i}\\ &#8211; 3\\overrightarrow{j} + 5\\overrightarrow{k}- (\\overrightarrow{i}\\ + \\overrightarrow{j}+\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=2\\overrightarrow{i}\\ &#8211; 3\\overrightarrow{j} + 5\\overrightarrow{k}- \\overrightarrow{i}\\ &#8211; \\overrightarrow{j}- \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{d}= \\overrightarrow{i}- 4\\overrightarrow{j}+4\\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Work\\  done =  \\overrightarrow{F}.\\overrightarrow{d}= (6\\overrightarrow{i}+ 10\\overrightarrow{j}-\\overrightarrow{k}) .(\\overrightarrow{i}- 4\\overrightarrow{j}+4\\overrightarrow{k})\\]<\/div>\n\n\n\n<p class=\"has-text-align-center\">=&nbsp; 6 ( 1 )&nbsp; + 10 ( -4 ) &#8211; 1 ( 4 )<\/p>\n\n\n\n<p class=\"has-text-align-center\">=&nbsp;&nbsp;&nbsp; 6&nbsp; &#8211; 40&nbsp; &#8211;&nbsp; 4<\/p>\n\n\n\n<p class=\"has-text-align-center\">=&nbsp;&nbsp;&nbsp; &#8211; 38<\/p>\n\n\n\n<p class=\"has-text-align-center\">Work done&nbsp; &nbsp;&nbsp;= &nbsp;&#8211;38 units<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Work\\ done\\ =\\ 38\\  units}\\ (by\\ taking\\ positive\\ value)\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/sxWdxbLemaU\" 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} {Example\\ 15\\ .}A\\ particle\\ is\\ displaced\\  from\\ the\\ point\\ 5\\overrightarrow{i}- 5\\overrightarrow{j}- 7\\overrightarrow{k}\\ to\\  the\\ point\\ 6\\overrightarrow{i}+ 2\\overrightarrow{j}-2\\overrightarrow{k}\\]\\[under\\ the\\ action\\ of\\ forces\\ 10\\overrightarrow{i}- \\overrightarrow{j} + 11\\overrightarrow{k},\\ 4\\overrightarrow{i} + 5\\overrightarrow{j} +6\\overrightarrow{k}\\ and\\ -2\\overrightarrow{i} + \\overrightarrow{j}- 9\\overrightarrow{k}.\\]\\[\\color {red} {Calculate\\ the\\ total\\ work\\ done\\ by\\ the\\ forces}.\\ \\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{F_1}= 10\\overrightarrow{i}- \\overrightarrow{j}+ 11\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F_2}= 4\\overrightarrow{i} + 5\\overrightarrow{j}+ 6\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F_3}= -2\\overrightarrow{i}+ \\overrightarrow{j}- 9\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F}=\\overrightarrow{F_1} + \\overrightarrow{F_2} + \\overrightarrow{F_3}\\]\\[= 10\\overrightarrow{i}- \\overrightarrow{j}+ 11\\overrightarrow{k} + 4\\overrightarrow{i} + 5\\overrightarrow{j}+ 6\\overrightarrow{k}- 2\\overrightarrow{i} + \\overrightarrow{j}- 9\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{F}= 12\\overrightarrow{i}+ 5\\overrightarrow{j} +8\\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}= 5\\overrightarrow{i}- 5\\overrightarrow{j}- 7\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}= 6\\overrightarrow{i} +2\\overrightarrow{j} &#8211; 2\\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} +2\\overrightarrow{j} &#8211; 2\\overrightarrow{k}- (5\\overrightarrow{i}- 5\\overrightarrow{j}- 7\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=6\\overrightarrow{i} +2\\overrightarrow{j} &#8211; 2\\overrightarrow{k}- 5\\overrightarrow{i} + 5\\overrightarrow{j}+ 7\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{d}= \\overrightarrow{i}+ 7\\overrightarrow{j}+5\\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Work\\  done =  \\overrightarrow{F}.\\overrightarrow{d}= (12\\overrightarrow{i}+ 5\\overrightarrow{j} +8\\overrightarrow{k}) .(\\overrightarrow{i}+ 7\\overrightarrow{j}+5\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=12(1) + 5(7) + 8(5)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= 12 + 35 + 40\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Work\\ done\\ = 87\\ units}\\]<\/div>\n\n\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<h3 class=\"wp-block-heading has-vivid-red-color has-text-color\">Vector Product of Two Vectors<\/h3>\n\n\n\n<h4 class=\"wp-block-heading has-vivid-cyan-blue-color has-text-color\">Definition<\/h4>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img data-recalc-dims=\"1\" decoding=\"async\" width=\"542\" height=\"407\" data-attachment-id=\"27716\" data-permalink=\"https:\/\/yanamtakshashila.com\/?attachment_id=27716\" data-orig-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/03\/image-12.png?fit=542%2C407&amp;ssl=1\" data-orig-size=\"542,407\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"image-12\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/03\/image-12.png?fit=542%2C407&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/03\/image-12.png?resize=542%2C407&#038;ssl=1\" alt=\"\" class=\"wp-image-27716\" srcset=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/03\/image-12.png?w=542&amp;ssl=1 542w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/03\/image-12.png?resize=486%2C365&amp;ssl=1 486w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/03\/image-12.png?resize=400%2C300&amp;ssl=1 400w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/03\/image-12.png?resize=200%2C150&amp;ssl=1 200w\" sizes=\"(max-width: 542px) 100vw, 542px\" \/><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let \\overrightarrow{a}\\ and  \\overrightarrow{b}\\  be\\ two\\ non\\ zero\\ vectors\\ inclined\\ at\\  an\\  angle\\ \\theta.\\] \\[Then\\ the\\ vector\\ product\\ of\\  \\overrightarrow{a} and \\overrightarrow{b} is\\  denoted\\  by\\]\\[\\overrightarrow{a}\u00d7\\overrightarrow{b} and\\ is\\ defined\\ as\\overrightarrow{a}\u00d7\\overrightarrow{b}=\\overrightarrow{|a|}\\overrightarrow{|b|}\\sin\\theta\\ n^\\wedge \\]<\/div>\n\n\n\n<p class=\"has-vivid-cyan-blue-color has-text-color\"><strong>Properties of Vector Product:<\/strong><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.    \\overrightarrow{a}\\ and\\overrightarrow{b}are\\ parellel\\ vectors\\  if\\ and\\ only\\ if  \\overrightarrow{a}\u00d7 \\overrightarrow{b}= 0. \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\   If \\overrightarrow{a}\\ and  \\overrightarrow{b}\\  are\\ any\\  two\\ vectors\\ then \\overrightarrow{a}\u00d7\\overrightarrow{b} = -\\overrightarrow{b}\u00d7\\overrightarrow{a}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.    \\overrightarrow{i},\\overrightarrow{j}\\  and\\  \\overrightarrow{k}\\ are\\  the\\ unit\\ vectors\\ along\\ the\\ x, y, z\\ axis\\  respectively   \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Then\\ i)  \\overrightarrow{i}\u00d7\\overrightarrow{i}= \\overrightarrow{j}\u00d7\\overrightarrow{j}=\\overrightarrow{k}\u00d7\\overrightarrow{k}=0  \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ ii)  \\overrightarrow{i}\u00d7\\overrightarrow{j}= \\overrightarrow{k};\\overrightarrow{j}\u00d7\\overrightarrow{k}= \\overrightarrow{i};\\overrightarrow{k}\u00d7\\overrightarrow{i}=\\overrightarrow{j}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ iii)  \\overrightarrow{j}\u00d7\\overrightarrow{i}= -\\overrightarrow{k};\\overrightarrow{k}\u00d7\\overrightarrow{j}= -\\overrightarrow{i};\\overrightarrow{i}\u00d7\\overrightarrow{k}= -\\overrightarrow{j}\\]<\/div>\n\n\n\n<p class=\"has-text-align-left\">                         4. Vector product in determinant form<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\  \\overrightarrow{a}= a_1\\overrightarrow{i}\\ + a_2\\overrightarrow{j}+ a_3\\overrightarrow{k}\\ and\\ \\overrightarrow{b}= b_1\\overrightarrow{i}\\ + b_2\\overrightarrow{j}+ b_3\\overrightarrow{k}\\ . Then\\]<\/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}\\\\\n\\overrightarrow{a_1} &amp; \\overrightarrow{a_2} &amp; \\overrightarrow{a_3}\\\\\n\\overrightarrow{b_1} &amp; \\overrightarrow{b_2} &amp; \\overrightarrow{b_3}\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5.\\   If \\overrightarrow{a}\\ and  \\overrightarrow{b}\\  are\\  two\\ adjacent\\ sides\\  of\\ a\\ parellelogram. Then\\ Area\\ of \\ parellelogram = |\\overrightarrow{a} \u00d7 \\overrightarrow{b}|\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[6.\\   If \\overrightarrow{d_1}\\ and  \\overrightarrow{d_2}\\  are\\  two\\ diagonals\\  of\\ a\\ parellelogram. Then\\ Area\\ of \\ parellelogram = \\frac{1}{2}|\\overrightarrow{d_1} \u00d7 \\overrightarrow{d_2}|\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[7.\\   If \\overrightarrow{a}\\ and  \\overrightarrow{b}\\  are\\  two\\ adjacent\\  sides\\ of\\ a\\ triangle.   Then\\ Area\\ of \\ triangle = \\frac{1}{2}|\\overrightarrow{a} \u00d7 \\overrightarrow{b}|\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[8.\\   Area\\ of\\ the\\ triangle\\ formed\\ by\\ the\\ points\\ whose\\ position\\ vectors\\  \\overrightarrow{OA},\\overrightarrow{OB}\\ and\\  \\overrightarrow{OC}\\ is\\   \\frac{1}{2}|\\overrightarrow{AB} \u00d7 \\overrightarrow{BC}|\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[9.\\  \\  sin\\  \\theta =\\frac{|\\overrightarrow{a} \u00d7 \\overrightarrow{b}|}{\\overrightarrow{|a|}\\overrightarrow{|b|}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[10.\\  n^\\wedge  =\\frac{\\overrightarrow{a}\u00d7 \\overrightarrow{b}}{|\\overrightarrow{a} \u00d7 \\overrightarrow{b}|}\\]<\/div>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Scalar triple Product<\/strong><\/h4>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ \\overrightarrow{a},\\overrightarrow{b} and\\  \\overrightarrow{c} be\\ any\\ three\\ vectors,\\]\\[their\\  scalar\\ triple\\ product\\ is\\ denoted\\ by\\ [\\overrightarrow{a}\\   \\overrightarrow{b}\\    \\overrightarrow{c}]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[11.\\ Let\\  \\overrightarrow{a}= a_1\\overrightarrow{i}\\ + a_2\\overrightarrow{j}+ a_3\\overrightarrow{k}\\ ,\\ \\overrightarrow{b}= b_1\\overrightarrow{i}\\ + b_2\\overrightarrow{j}+ b_3\\overrightarrow{k} and\\ \\overrightarrow{c}= c_1\\overrightarrow{i}\\ + c_2\\overrightarrow{j}+ c_3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Then\\ [\\overrightarrow{a}\\   \\overrightarrow{b}    \\overrightarrow{c}] =\\begin{vmatrix}\na_1 &amp; a_2 &amp; a_3\\\\\nb_1 &amp; b_2 &amp; b_3\\\\\nc_1 &amp; c_2 &amp; c_3\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\underline{\\color {red} {12.\\ Condition\\ for\\ three\\ vectors\\ \\overrightarrow{a}\\, \\overrightarrow{b}\\ and\\ \\overrightarrow{c}\\ to\\ be\\ coplanar\\ (or)\\ lie\\ on\\ the\\ same\\ plane}}\\]\\[if\\ \\color {green} {\\begin{vmatrix}\n{a_1} &amp;{a_2} &amp; {a_3}\\\\\n{b_1} &amp; {b_2} &amp; {b_3}\\\\\n{c_1} &amp; {c_2} &amp; {c_3}\\\\\n\\end{vmatrix}\\ =\\ 0}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 16\\ .}\\ \\color {red} {Find\\  \\overrightarrow{a}\u00d7 \\overrightarrow{b}}\\ if\\  \\overrightarrow{a}= \\overrightarrow{i}+ \\overrightarrow{j}+ \\overrightarrow{k} and \\overrightarrow{b}= 2\\overrightarrow{i}- \\overrightarrow{j}\\ +\\ \\overrightarrow{k}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{18cm}\\ April\\ 2024\\]<\/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}= \\overrightarrow{i}+ \\overrightarrow{j}+\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= 2\\overrightarrow{i}\\ -\\ \\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; 1\\\\\n2 &amp; -1 &amp; 1\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( 1 + 1)\\ -\\ \\overrightarrow{j}(1\\ -\\ 2)\\ +\\ \\overrightarrow{k}(-1\\ -\\ 2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(2)\\ -\\ \\overrightarrow{j}(-\\ 1)\\ +\\ \\overrightarrow{k}(-3)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{ \\overrightarrow{a}\u00d7 \\overrightarrow{b}\\ =\\ 2\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -3\\overrightarrow{k}}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/nd7f3fG4hIQ\" title=\"Product of Vectors - 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\">\\[\\color {purple} {Example\\ 17\\ .}\\ \\color {red} {Show\\ that}\\ \\overrightarrow{i}\\ -\\  2\\overrightarrow{j}\\ -\\ 4\\overrightarrow{k}\\ and\\ 3\\overrightarrow{i}\\ -\\ 6\\overrightarrow{j}\\ -\\ 12\\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}\\ =\\ \\overrightarrow{i}\\ -\\  2\\overrightarrow{j}\\ -\\ 4\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}\\ =\\ 3\\overrightarrow{I}\\ -\\ 6\\overrightarrow{j}\\ -\\ 12\\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; -2 &amp; -4\\\\\n3 &amp; -6 &amp; -12\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( 24\\ -\\ 24)\\  -\\overrightarrow{j}(-12\\ +\\ 12)\\ +\\ \\overrightarrow{k}(-6\\ +\\ 6)\\]<\/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\/EKWTwXmfPrM\" title=\"Product of Vectors - 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\">\\[\\color {purple} {Example\\ 18\\ .}\\ \\color {red} {Find\\ the\\ area\\ of\\ the\\ parellelogram}\\ whose\\ adjacent\\ sides\\ are\\ \\hspace{10cm}\\]\\[-\\ \\overrightarrow{i}\\ + 2\\overrightarrow{j}\\ +\\  4\\overrightarrow{k}\\ and\\  \\overrightarrow{i}\\ &#8211; \\overrightarrow{j}\\ -\\ \\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{a}=-\\ \\overrightarrow{i}\\ + 2\\overrightarrow{j}\\ +\\  4\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= \\overrightarrow{i}\\ &#8211; \\overrightarrow{j}\\ -\\ \\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\">\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/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}\\\\\n&#8211; 1 &amp;  2 &amp; 4\\\\\n1 &amp; -1 &amp; -1\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(-\\ 2 +\\ 4)\\ -\\ \\overrightarrow{j}(1\\ -\\ 4)\\ +\\ \\overrightarrow{k}(1\\ -\\ 2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(2)\\ -\\overrightarrow{j}(-\\ 3)\\ +\\ \\overrightarrow{k}(- 1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\overrightarrow{a}\u00d7 \\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}| =  \\sqrt{(2)^2\\ +\\ (3)^2 + (-1)^2 }=\\sqrt{(4\\ +\\ 9\\ +\\ 1}=\\sqrt{14}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Area\\ of \\ parellelogram = \\sqrt{14} sq.units}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/_OMrpTsDN6c\" title=\"Product of Vectors - 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\">\\[\\color {purple} {Example\\ 19\\ .}\\ \\color {red} {Find\\ the\\ area\\ of\\ the\\ parellelogram}\\ whose\\ diagonals\\ are\\ represented\\ by\\ \\hspace{10cm}\\]\\[3\\ \\overrightarrow{i}\\ + \\overrightarrow{j}\\ -\\  2\\overrightarrow{k}\\ and\\  \\overrightarrow{i}\\ -\\ 3\\ \\overrightarrow{j}\\ +\\ 4\\ \\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{d_1}\\ =\\ 3\\ \\overrightarrow{i}\\ + \\overrightarrow{j}\\ -\\  2\\overrightarrow{k}\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{d_2}\\ =\\ \\overrightarrow{i}\\ -\\ 3\\ \\overrightarrow{j}\\ +\\ 4\\ \\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}\\\\\n3 &amp;  1 &amp; -2\\\\\n1 &amp; -3 &amp; 4\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( 4\\ -\\ 6) -\\overrightarrow{j}(12\\ +\\ 2)+\\overrightarrow{k}(-9\\ -\\ 1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(-2) -\\overrightarrow{j}(14)\\ +\\ \\overrightarrow{k}(- 10)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{d_1}\\ \u00d7\\  \\overrightarrow{d_2}\\ =\\ -\\ 2\\overrightarrow{i}\\  -\\ 14\\ \\overrightarrow{j}\\ -\\ 10\\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{(-2)^2 + (-14)^2 + (-10)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  \\frac{1}{2}\\ \\sqrt{(4\\ +\\  196\\ +\\ 100)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{2}\\ \\sqrt{300}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Area\\ of \\ parellelogram =\\ \\frac{1}{2}\\ \\sqrt{300} sq.units}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/Iccvs8Z6yWM\" title=\"Product of vectors - 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=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 20\\ .}\\ \\color {red} {Find\\ the\\ area\\ of\\ the\\ triangle}\\ formed\\ by\\ the\\ points\\ whose\\ position\\ vectors\\ \\hspace{8cm}\\]\\[   3\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ -\\overrightarrow{k}\\ ,  2\\overrightarrow{i}\\ -\\ 3\\overrightarrow{j}\\ +\\ \\overrightarrow{k}\\ and\\  5\\overrightarrow{i}\\ + \\overrightarrow{j}\\ +\\ 3\\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{OA}= 3\\overrightarrow{i}\\ + 2\\overrightarrow{j}- \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}= 2\\overrightarrow{i}\\ -3\\overrightarrow{j} +\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OC}= 5\\overrightarrow{i}\\ +\\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\">\\[=2\\overrightarrow{i}\\ -3\\overrightarrow{j} +\\overrightarrow{k}- (3\\overrightarrow{i} + 2\\overrightarrow{j}- \\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=2\\overrightarrow{i}\\ -3\\overrightarrow{j} +\\overrightarrow{k}- 3\\overrightarrow{i}- 2\\overrightarrow{j}+ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{AB}=  -\\overrightarrow{i} &#8211; 5\\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\">\\[=5\\overrightarrow{i}\\ +\\overrightarrow{j} + 3\\overrightarrow{k}- (2\\overrightarrow{i} -3\\overrightarrow{j} +\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=5\\overrightarrow{i}\\ +\\overrightarrow{j} + 3\\overrightarrow{k}- 2\\overrightarrow{i} + 3\\overrightarrow{j} -\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{BC}=  3\\overrightarrow{i} + 4\\overrightarrow{j} + 2\\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-1 &amp; -5 &amp; 2\\\\\n3 &amp; 4 &amp; 2\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( -10 &#8211; 8) -\\overrightarrow{j}(-2 &#8211; 6)+\\overrightarrow{k}(-4 + 15)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(-18) -\\overrightarrow{j}(8)+\\overrightarrow{k}(11)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{AB}\u00d7 \\overrightarrow{BC}= -18\\overrightarrow{i}\\ +\\ 8\\overrightarrow{j}+11\\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[|\\overrightarrow{AB} \u00d7 \\overrightarrow{BC}| =  \\sqrt{(-18 )^2 + (-8)^2 + (11)^2 }=\\sqrt{(324 + 64 + 121 }=\\sqrt{509}\\]<\/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{509}}{2}\\ sq. units}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/7wDM756pP84\" title=\"Product of Vectors - 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.\\ \\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\">\\[\\hspace{18cm}\\ April\\ 2024\\]<\/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}\\ =\\ 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\">\\[\\color {purple} {Example\\ 22\\ .}\\ If\\ |\\overrightarrow{a}| = 2,\\ |\\overrightarrow{b}|= 7\\ and\\ |\\overrightarrow{a} \u00d7 \\overrightarrow{b}|=7,\\ \\color {red} {find\\ the\\ angle\\ between\\ \\overrightarrow{a}\\ and\\ \\overrightarrow{b}}.\\ \\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\">\\[\\  sin\\  \\theta =\\frac{|\\overrightarrow{a} \u00d7 \\overrightarrow{b}|}{\\overrightarrow{|a|}\\overrightarrow{|b|}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\frac{7}{(2)(7)} = \\frac{1}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\theta = 30^{\\circ}}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/Uhw5Qp4T7wY\" title=\"Product of Vectors - 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\">\\[\\color {purple} {Example\\ 23\\ .}\\ \\color {red} {Find\\ the\\ unit\\ vector}\\ perpendicular\\ to\\ each\\ of\\ the\\ vectors\\   2\\overrightarrow{i} -\\overrightarrow{j}+\\overrightarrow{k}   and\\  3\\overrightarrow{i}+ 4\\overrightarrow{j} -\\overrightarrow{k}.\\]\\[\\color {red} {Also\\ find\\ the\\ sine\\ of\\ the\\ angle}\\ between\\ the\\ vectors .\\]<\/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\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}= 2\\overrightarrow{i} -\\overrightarrow{j}+\\overrightarrow{k} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}=3\\overrightarrow{i}+ 4\\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; -1 &amp; 1\\\\\n3 &amp;  4 &amp; -1\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( 1 &#8211; 4) -\\overrightarrow{j}(-2 &#8211; 3)+\\overrightarrow{k}(8 + 3)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(-3) -\\overrightarrow{j}(-5)+\\overrightarrow{k}(11)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{a}\u00d7 \\overrightarrow{b}= -3\\overrightarrow{i} +5\\overrightarrow{j}+11\\overrightarrow{k}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[|\\overrightarrow{a} \u00d7 \\overrightarrow{b}| =  \\sqrt{(-3 )^2 + (5)^2 + (11)^2 }=\\sqrt{9 + 25 + 121}=\\sqrt{155}\\]<\/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{-3\\overrightarrow{i}\\ + 5\\overrightarrow{j}+ 11\\overrightarrow{k}}{\\sqrt{155}} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{n^\\wedge  = \\frac{-3\\overrightarrow{i}\\ + 5\\overrightarrow{j}+ 11\\overrightarrow{k}}{\\sqrt{155}}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{|a|} = \\sqrt{(2)^2 + (-1)^2 + (1)^2 }=\\sqrt{(4 + 1 + 1 }=\\sqrt{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{|b|} = \\sqrt{(3)^2 + (4)^2 + (-1)^2 }=\\sqrt{(9 + 16 + 1 }=\\sqrt{26}\\]<\/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{155}}{\\sqrt{6}\\sqrt{26}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{sin\\  \\theta =\\frac{\\sqrt{155}}{\\sqrt{6}\\sqrt{26}}}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/UOZS2t6Vr7s\" title=\"Product of Vectors - 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\"><\/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=\"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<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<h3 class=\"wp-block-heading has-vivid-red-color has-text-color\">Application of Vector Product<\/h3>\n\n\n\n<p class=\"has-vivid-cyan-blue-color has-text-color has-medium-font-size\"><strong>Moment (or) Torque of a force about a point<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img data-recalc-dims=\"1\" decoding=\"async\" width=\"480\" height=\"250\" data-attachment-id=\"27862\" data-permalink=\"https:\/\/yanamtakshashila.com\/?attachment_id=27862\" data-orig-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/03\/image-22.png?fit=480%2C250&amp;ssl=1\" data-orig-size=\"480,250\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"image-22\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/03\/image-22.png?fit=480%2C250&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/03\/image-22.png?resize=480%2C250&#038;ssl=1\" alt=\"\" class=\"wp-image-27862\"\/><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ A\\ be\\ any\\ point\\ and\\ \\overrightarrow{r}\\ be\\ the\\ position\\ vector\\ relative\\ to\\ the\\ point\\ A\\]\\[of\\ any\\ point\\ P\\ on\\ the\\ line\\ of\\ the\\ action\\ of\\ the\\ force\\ \\overrightarrow{F}.\\]\\[The\\ moment\\ of\\ the\\ force\\ about\\ the\\ point\\ A\\ is\\ defined\\ as\\ \\overrightarrow{M}= \\overrightarrow{r}\u00d7 \\overrightarrow{F}\\]\\[where\\  \\overrightarrow{r}= \\overrightarrow{AP}= \\overrightarrow{OP}- \\overrightarrow{OA}\\].<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ The\\ Magnitude \\ of\\ Moment = |\\overrightarrow{r} \u00d7 \\overrightarrow{F}|\\]<\/div>\n\n\n\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\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 24\\ .}\\ \\color {red} {Find\\ the\\ moment\\ of\\ the\\ force}\\ 3\\overrightarrow{i}\\ +\\ \\overrightarrow{k}\\ \\hspace{8cm}\\]\\[acting\\ through\\ the\\ point\\ \\overrightarrow{i}+2\\overrightarrow{j}-\\overrightarrow{k}\\ about\\ the\\ point\\  2\\overrightarrow{i}+ \\overrightarrow{j}-2\\overrightarrow{k}.\\]<\/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}= 3\\overrightarrow{i} + \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OP}= \\overrightarrow{i} +2\\overrightarrow{j} -\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}= 2\\overrightarrow{i} +\\overrightarrow{j} -2\\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} +2\\overrightarrow{j} -\\overrightarrow{k}- (2\\overrightarrow{i} +\\overrightarrow{j} -2\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\overrightarrow{i} +2\\overrightarrow{j} -\\overrightarrow{k}- 2\\overrightarrow{i}-\\overrightarrow{j} +2\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{r}=  -\\overrightarrow{i} + \\overrightarrow{j} + \\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-1 &amp; 1 &amp; 1\\\\\n3 &amp; 0 &amp; 1\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( 1 &#8211; 0) -\\overrightarrow{j}(-1 &#8211; 3)+\\overrightarrow{k}(0 &#8211; 3)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}(1) -\\overrightarrow{j}(-4)+\\overrightarrow{k}(-3)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{r}\u00d7 \\overrightarrow{F}= \\overrightarrow{i}+ 4\\overrightarrow{j}-3\\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{(1)^2 + (4)^2 + (-3)^2 }=\\sqrt{(1 + 16 +9 }=\\sqrt{26}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Magnitude\\ of \\ Moment = \\sqrt{26}\\ units}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/bfUGgmhcaKQ\" title=\"Product of Vectors - 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\">\\[\\color {purple} {Example\\ 25\\ .}\\ \\color {red} {Find\\ the\\ Magnitude\\ of\\ the\\ moment\\ of\\ the\\ force}\\ 6\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -\\ \\overrightarrow{k}\\ \\hspace{7cm}\\]\\[acting\\ along\\ the\\ point\\ (0,\\ 1,\\ -\\ 1)\\ about\\ the\\ point\\ (4,\\ 3,\\ -\\ 1)\\ \\hspace{8cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{18cm}\\ April\\ 2024\\]<\/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\">\\[\\color {blue} {Soln:}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{F}= 6\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -\\ \\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{OP}\\ =\\ 0\\overrightarrow{i}\\ + \\overrightarrow{j}\\ -\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OP}\\ =\\  \\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{j}\\ -\\ \\overrightarrow{k}\\ &#8211; (4\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ -\\ \\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\overrightarrow{j}\\ -\\ \\overrightarrow{k}\\ -\\  4\\overrightarrow{i}\\ -\\ 3\\overrightarrow{j}\\ +\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{r}\\ =\\  -\\ 4\\overrightarrow{i}\\ -\\ 2\\overrightarrow{j}}\\]<\/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-4 &amp; -2 &amp; 0\\\\\n6 &amp; 1 &amp; &#8211; 1\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\overrightarrow{i}( 2 &#8211; 0) -\\overrightarrow{j}(4 &#8211; 0)\\ +\\ \\overrightarrow{k}(-4 + 12)\\]<\/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{(2)^2 + (-4)^2 + (8)^2 }=\\sqrt{(4\\ +\\ 16\\ +\\ 64 }\\ =\\sqrt{84}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Magnitude\\ of \\ Moment = \\sqrt{84}}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/JIbANpbqxpQ\" title=\"Product of Vectors - 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<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- Leader board 1 -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:728px;height:90px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8769628924\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 26\\ .}\\ \\color {red} {Find\\  the\\  value\\  of}\\ [\\overrightarrow{i} + \\overrightarrow{j}\\ \\overrightarrow{j} + \\overrightarrow{k}\\ \\overrightarrow{k} + \\overrightarrow{i}]\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{18cm}\\ April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{18cm}\\]<\/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}= \\overrightarrow{i}\\ + \\overrightarrow{j}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= \\overrightarrow{j}\\ + \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{c}= \\overrightarrow{k} + \\overrightarrow{i}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ [\\overrightarrow{a}\\   \\overrightarrow{b}    \\overrightarrow{c}] =\\begin{vmatrix}\n1 &amp; 1 &amp; 0\\\\\n0 &amp; 1 &amp; 1\\\\\n1 &amp; 0 &amp; 1\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 1(1\\ -\\ 0)\\ -\\ 1(0\\ -\\ 1)\\ +\\ 0(0\\ -\\ 1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 1(1)\\ -\\ 1(-1)\\ +\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 1\\ +\\ 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 27\\ .}\\ \\color {red} {Show\\ that}\\ 5\\overrightarrow{i}\\ +\\  6\\overrightarrow{j}\\ +\\ 7\\overrightarrow{k},\\ 7\\overrightarrow{i}\\ -\\  8\\overrightarrow{j}\\ +\\ 9\\overrightarrow{k}\\ and\\ 3\\overrightarrow{i}\\ +\\ 20\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k}\\ \\color {red} {are\\ coplanar}.\\ \\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}\\ =\\ 5\\overrightarrow{i}\\ +\\  6\\overrightarrow{j}\\ +\\ 7\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}\\ =\\ 7\\overrightarrow{i}\\ -\\  8\\overrightarrow{j}\\ +\\ 9\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{c}\\ =\\ 3\\overrightarrow{i}\\ +\\ 20\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ [\\overrightarrow{a}\\   \\overrightarrow{b}    \\overrightarrow{c}] =\\begin{vmatrix}\n5 &amp; 6 &amp; 7\\\\\n7 &amp; -8 &amp; 9\\\\\n3 &amp; 20 &amp; 5\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 5(-\\ 40\\ -\\ 180)\\ -\\ 6(35\\ -\\ 27)\\ +\\ 7(140\\ +\\ 24)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 5(-220)\\ -\\ 6(8)\\ +\\ 7(164)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ -\\ 1100\\ -\\ 48\\ +\\ 1148\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 0\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/pYMyPUo9Jgc\" title=\"Coplanar ( Three vectors)\" 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} {Example\\ 28\\ .}\\ \\color {red} {Find\\ the\\ value\\ of\\ &#8216;m&#8217;\\ so\\ that\\ the\\ vectors}\\  \\hspace{10cm}\\]\\[2\\ \\overrightarrow{i}\\ &#8211; \\overrightarrow{j}\\ +\\  \\overrightarrow{k},\\ \\overrightarrow{i}\\ +\\ 2\\ \\overrightarrow{j}\\ -\\ 3\\ \\overrightarrow{k}\\ ,\\ 3\\overrightarrow{i}\\ +\\ m\\ \\overrightarrow{j}\\ +\\ 5\\ \\overrightarrow{k}\\ coplanar\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{18cm}\\ April\\ 2024\\]<\/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}+ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}= \\overrightarrow{i}+ 2\\overrightarrow{j}- 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{c}= 3\\overrightarrow{i}+ m\\overrightarrow{j}+ 5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\  \\overrightarrow{a},\\overrightarrow{b} and\\  \\overrightarrow{c} are\\ coplanar\\ \\implies  [\\overrightarrow{a}\\   \\overrightarrow{b}\\    \\overrightarrow{c}] = 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\n2 &amp;- 1 &amp; 1\\\\\n1 &amp; 2 &amp; -3\\\\\n3 &amp; m &amp; 5\\\\\n\\end{vmatrix}=0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2(10\\ +\\ 3\\ m)\\ +\\ 1(5\\ +\\ 9)\\ +\\ 1(m\\ -\\ 6)\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[20\\ +\\ 6m\\ +\\ 14\\ +\\ m\\ -\\ 6\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[28\\ +\\ 7m\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[7m\\ =\\ -\\ 28\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[m\\ =\\ -\\ 4\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 29\\ .}\\ \\color {red} {Find\\ the\\ value\\ of\\ &#8216;p&#8217;\\ such\\ that\\ the\\ vectors}\\  \\hspace{10cm}\\]\\[2\\ \\overrightarrow{i}\\ -\\ 3\\ \\overrightarrow{j}\\ +\\  5\\ \\overrightarrow{k},\\ p\\ \\overrightarrow{i}\\ +\\ 2\\ \\overrightarrow{j}\\ -\\ \\ \\overrightarrow{k}\\ and\\ \\ 3\\overrightarrow{i}\\ -\\  \\overrightarrow{j}\\ +\\ 4\\ \\overrightarrow{k}\\ lie\\ on\\ the\\ same\\ plane\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{18cm}\\ April\\ 2024\\]<\/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}\\ -\\ 3\\ \\overrightarrow{j}\\ +\\  5\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}\\ =\\ p\\ \\overrightarrow{i}\\ +\\ 2\\ \\overrightarrow{j}\\ -\\ \\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{c}\\ =\\ 3\\overrightarrow{i}\\ -\\  \\overrightarrow{j}\\ +\\ 4\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\  \\overrightarrow{a},\\overrightarrow{b} and\\  \\overrightarrow{c}\\ lie\\ on\\ the\\ same\\ plane\\  \\implies\\ [\\overrightarrow{a}\\   \\overrightarrow{b}\\    \\overrightarrow{c}] = 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\n2 &amp;- 3 &amp; 5\\\\\np &amp; 2 &amp; -1\\\\\n3 &amp; -1 &amp; 4\\\\\n\\end{vmatrix}=0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2(8\\ -\\ 1)\\ +\\ 3(4\\ p\\ +\\ 3)\\ +\\ 5(-\\ p\\ -\\ 6)\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[14\\ +\\ 12\\ p\\ +\\ 9\\ +\\ m\\ -\\ 5\\ p\\ -\\ 30\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[-\\ 7\\ +\\ 7\\ p\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[7\\ p\\ =\\  7\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[p\\ =\\ 1\\]<\/div>\n","protected":false},"excerpt":{"rendered":"<p>SCALAR PRODUCT Definition:&nbsp; Properties of Scalar Product: = 2(1) &#8211; 4 (6) + 8 (12) = 2 &#8211; 24 + 96 = 74 3(-6) &#8211; 1(m) + 5 (4) = 0 -18 &#8211; m + 20 = 0 -m + 2 = 0 = 1(0) &#8211; 1(4) + 2 (2) = 0 = 0(-10) + [&hellip;]<\/p>\n","protected":false},"author":187055548,"featured_media":0,"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 center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"_wpas_customize_per_network":false,"jetpack_post_was_ever_published":false},"categories":[711787869],"tags":[],"class_list":["post-12526","post","type-post","status-publish","format-standard","hentry","category-product-of-two-vectors"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>2.2 PRODUCT OF VECTORS - 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=12526\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"2.2 PRODUCT OF VECTORS - YANAMTAKSHASHILA\" \/>\n<meta property=\"og:description\" content=\"SCALAR PRODUCT Definition:&nbsp; Properties of Scalar Product: = 2(1) &#8211; 4 (6) + 8 (12) = 2 &#8211; 24 + 96 = 74 3(-6) &#8211; 1(m) + 5 (4) = 0 -18 &#8211; m + 20 = 0 -m + 2 = 0 = 1(0) &#8211; 1(4) + 2 (2) = 0 = 0(-10) + [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/yanamtakshashila.com\/?p=12526\" \/>\n<meta property=\"og:site_name\" content=\"YANAMTAKSHASHILA\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/profile.php?id=100063680185552\" \/>\n<meta property=\"article:author\" content=\"https:\/\/www.facebook.com\/profile.php?id=100063680185552\" \/>\n<meta property=\"article:published_time\" content=\"2021-03-01T11:17:06+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-04-20T06:03:33+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/files.askiitians.com\/cdn1\/images\/2014108-144310695-8606-dot-product-image.png\" \/>\n<meta name=\"author\" content=\"rajuviswa\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rajuviswa\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"8 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12526#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12526\"},\"author\":{\"name\":\"rajuviswa\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\"},\"headline\":\"2.2 PRODUCT OF VECTORS\",\"datePublished\":\"2021-03-01T11:17:06+00:00\",\"dateModified\":\"2024-04-20T06:03:33+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12526\"},\"wordCount\":4568,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\"},\"image\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12526#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/files.askiitians.com\\\/cdn1\\\/images\\\/2014108-144310695-8606-dot-product-image.png\",\"articleSection\":[\"Product of two vectors\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12526#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12526\",\"url\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12526\",\"name\":\"2.2 PRODUCT OF VECTORS - YANAMTAKSHASHILA\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12526#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12526#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/files.askiitians.com\\\/cdn1\\\/images\\\/2014108-144310695-8606-dot-product-image.png\",\"datePublished\":\"2021-03-01T11:17:06+00:00\",\"dateModified\":\"2024-04-20T06:03:33+00:00\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12526#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12526\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12526#primaryimage\",\"url\":\"https:\\\/\\\/files.askiitians.com\\\/cdn1\\\/images\\\/2014108-144310695-8606-dot-product-image.png\",\"contentUrl\":\"https:\\\/\\\/files.askiitians.com\\\/cdn1\\\/images\\\/2014108-144310695-8606-dot-product-image.png\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12526#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/yanamtakshashila.com\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"2.2 PRODUCT OF VECTORS\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#website\",\"url\":\"https:\\\/\\\/yanamtakshashila.com\\\/\",\"name\":\"yanamtakshashila.com\",\"description\":\"one stop solutions\",\"publisher\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/yanamtakshashila.com\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":[\"Person\",\"Organization\"],\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\",\"name\":\"rajuviswa\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2024\\\/12\\\/LOGO-PNG.png?fit=3600%2C3600&ssl=1\",\"url\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2024\\\/12\\\/LOGO-PNG.png?fit=3600%2C3600&ssl=1\",\"contentUrl\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2024\\\/12\\\/LOGO-PNG.png?fit=3600%2C3600&ssl=1\",\"width\":3600,\"height\":3600,\"caption\":\"rajuviswa\"},\"logo\":{\"@id\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2024\\\/12\\\/LOGO-PNG.png?fit=3600%2C3600&ssl=1\"},\"sameAs\":[\"http:\\\/\\\/yanamtakshashila.wordpress.com\",\"https:\\\/\\\/www.facebook.com\\\/profile.php?id=100063680185552\",\"https:\\\/\\\/www.instagram.com\\\/rajuviswa\\\/?hl=en\",\"https:\\\/\\\/www.youtube.com\\\/channel\\\/UCjJ2KWWvsFm6F42UtMdbxzw\"],\"url\":\"https:\\\/\\\/yanamtakshashila.com\\\/?author=187055548\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"2.2 PRODUCT OF VECTORS - YANAMTAKSHASHILA","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:\/\/yanamtakshashila.com\/?p=12526","og_locale":"en_US","og_type":"article","og_title":"2.2 PRODUCT OF VECTORS - YANAMTAKSHASHILA","og_description":"SCALAR PRODUCT Definition:&nbsp; Properties of Scalar Product: = 2(1) &#8211; 4 (6) + 8 (12) = 2 &#8211; 24 + 96 = 74 3(-6) &#8211; 1(m) + 5 (4) = 0 -18 &#8211; m + 20 = 0 -m + 2 = 0 = 1(0) &#8211; 1(4) + 2 (2) = 0 = 0(-10) + [&hellip;]","og_url":"https:\/\/yanamtakshashila.com\/?p=12526","og_site_name":"YANAMTAKSHASHILA","article_publisher":"https:\/\/www.facebook.com\/profile.php?id=100063680185552","article_author":"https:\/\/www.facebook.com\/profile.php?id=100063680185552","article_published_time":"2021-03-01T11:17:06+00:00","article_modified_time":"2024-04-20T06:03:33+00:00","og_image":[{"url":"https:\/\/files.askiitians.com\/cdn1\/images\/2014108-144310695-8606-dot-product-image.png","type":"","width":"","height":""}],"author":"rajuviswa","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rajuviswa","Est. reading time":"8 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/yanamtakshashila.com\/?p=12526#article","isPartOf":{"@id":"https:\/\/yanamtakshashila.com\/?p=12526"},"author":{"name":"rajuviswa","@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e"},"headline":"2.2 PRODUCT OF VECTORS","datePublished":"2021-03-01T11:17:06+00:00","dateModified":"2024-04-20T06:03:33+00:00","mainEntityOfPage":{"@id":"https:\/\/yanamtakshashila.com\/?p=12526"},"wordCount":4568,"commentCount":0,"publisher":{"@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e"},"image":{"@id":"https:\/\/yanamtakshashila.com\/?p=12526#primaryimage"},"thumbnailUrl":"https:\/\/files.askiitians.com\/cdn1\/images\/2014108-144310695-8606-dot-product-image.png","articleSection":["Product of two vectors"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/yanamtakshashila.com\/?p=12526#respond"]}]},{"@type":"WebPage","@id":"https:\/\/yanamtakshashila.com\/?p=12526","url":"https:\/\/yanamtakshashila.com\/?p=12526","name":"2.2 PRODUCT OF VECTORS - YANAMTAKSHASHILA","isPartOf":{"@id":"https:\/\/yanamtakshashila.com\/#website"},"primaryImageOfPage":{"@id":"https:\/\/yanamtakshashila.com\/?p=12526#primaryimage"},"image":{"@id":"https:\/\/yanamtakshashila.com\/?p=12526#primaryimage"},"thumbnailUrl":"https:\/\/files.askiitians.com\/cdn1\/images\/2014108-144310695-8606-dot-product-image.png","datePublished":"2021-03-01T11:17:06+00:00","dateModified":"2024-04-20T06:03:33+00:00","breadcrumb":{"@id":"https:\/\/yanamtakshashila.com\/?p=12526#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/yanamtakshashila.com\/?p=12526"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/yanamtakshashila.com\/?p=12526#primaryimage","url":"https:\/\/files.askiitians.com\/cdn1\/images\/2014108-144310695-8606-dot-product-image.png","contentUrl":"https:\/\/files.askiitians.com\/cdn1\/images\/2014108-144310695-8606-dot-product-image.png"},{"@type":"BreadcrumbList","@id":"https:\/\/yanamtakshashila.com\/?p=12526#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/yanamtakshashila.com\/"},{"@type":"ListItem","position":2,"name":"2.2 PRODUCT OF VECTORS"}]},{"@type":"WebSite","@id":"https:\/\/yanamtakshashila.com\/#website","url":"https:\/\/yanamtakshashila.com\/","name":"yanamtakshashila.com","description":"one stop solutions","publisher":{"@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/yanamtakshashila.com\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":["Person","Organization"],"@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e","name":"rajuviswa","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/12\/LOGO-PNG.png?fit=3600%2C3600&ssl=1","url":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/12\/LOGO-PNG.png?fit=3600%2C3600&ssl=1","contentUrl":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/12\/LOGO-PNG.png?fit=3600%2C3600&ssl=1","width":3600,"height":3600,"caption":"rajuviswa"},"logo":{"@id":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/12\/LOGO-PNG.png?fit=3600%2C3600&ssl=1"},"sameAs":["http:\/\/yanamtakshashila.wordpress.com","https:\/\/www.facebook.com\/profile.php?id=100063680185552","https:\/\/www.instagram.com\/rajuviswa\/?hl=en","https:\/\/www.youtube.com\/channel\/UCjJ2KWWvsFm6F42UtMdbxzw"],"url":"https:\/\/yanamtakshashila.com\/?author=187055548"}]}},"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_likes_enabled":true,"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/pc3kmt-3g2","_links":{"self":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/12526","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/users\/187055548"}],"replies":[{"embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=12526"}],"version-history":[{"count":100,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/12526\/revisions"}],"predecessor-version":[{"id":45912,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/12526\/revisions\/45912"}],"wp:attachment":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=12526"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=12526"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=12526"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}