{"id":12376,"date":"2021-02-27T15:42:35","date_gmt":"2021-02-27T10:12:35","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=12376"},"modified":"2024-04-16T16:12:10","modified_gmt":"2024-04-16T10:42:10","slug":"2-1-vector-introduction","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=12376","title":{"rendered":"2.1 VECTOR \u2013 INTRODUCTION"},"content":{"rendered":"\n<p class=\"has-text-align-justify\">Vectors constitute one of the several Mathematical systems which can be usefully employed to provide mathematical handling for certain types of problems in Geometry, Mechanics and other branches of Applied Mathematics. Vectors facilitate mathematical study of such physical quantities as possess Direction in addition to Magnitude. Velocity of a particle, for example, is one such quantity.<\/p>\n\n\n\n<div class=\"aicp\"><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<p>Physical quantities are broadly divided in two categories viz (a) <strong>Vector Quantities<\/strong>  &amp; (b) <strong>Scalar quantities<\/strong>.<\/p>\n\n\n\n<p>( a ) <strong>Vector quantities<\/strong> :<\/p>\n\n\n\n<p>Any quantity, such as velocity, momentum, or force, that has both magnitude and direction<\/p>\n\n\n\n<p>( b ) <strong>Scalar quantities<\/strong> :<\/p>\n\n\n\n<p>A quantity, such as mass, length, time, density or energy, that has size or magnitude but does not involve the concept of direction is called scalar quantity.<\/p>\n\n\n\n<div class=\"aicp\"><script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block; text-align:center;\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"2812384453\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n<\/div>\n\n\n\n<p><strong>Mathematical Description of Vector:<\/strong> A vector is a directed line segment. The length of the segment is called magnitude of the vector. The direction is indicated by an arrow joining the initial and final points of the line segment. The vector AB i.e, joining the initial point A and the final point B in the direction of AB is denoted as <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}\\]<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\"><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\\ 1\\ .}\\ \\overrightarrow{a}\\ =\\ 2\\overrightarrow{i}\\ +\\  3\\overrightarrow{j}\\ -\\  \\overrightarrow{k}\\ and\\  \\overrightarrow{b}\\ =\\ \\overrightarrow{j}\\ &#8211; 2\\ \\overrightarrow{k},\\ \\color {red} {find\\   \\overrightarrow{a}\\ -\\   2\\overrightarrow{b}}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ Given\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\\ =\\ 2\\overrightarrow{i}\\ +\\  3\\overrightarrow{j}\\ -\\  \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}\\ =\\ \\overrightarrow{j}\\ &#8211; 2\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\\ -\\   2\\overrightarrow{b}\\ =\\ 3\\overrightarrow{i}\\ + 2\\overrightarrow{j} + \\overrightarrow{k}\\ -\\  2(\\overrightarrow{j}\\ -\\ 2\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 3\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ +\\  \\overrightarrow{k}\\ -\\  2\\overrightarrow{j}\\ +\\ 4\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\\ -\\   2\\overrightarrow{b}\\ =\\ 3\\overrightarrow{i}\\ +\\  5\\overrightarrow{k}\\ \\hspace{5cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/cO-Vo2gG7O4\" title=\"Vector Introduction - Part - 1\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<p class=\"has-text-align-justify\">   <strong>Triangle Law of&nbsp; Addition of Two vectors:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img data-recalc-dims=\"1\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.studypage.in\/images\/maths\/algebra\/triangle-law-vector-add.png?ssl=1\" alt=\"Addition of Vectors - Study Page\" style=\"width:466px;height:165px\"\/><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA} = \\overrightarrow{a}\\] \\[\\overrightarrow{AB} = \\overrightarrow{b}\\]\\[\\overrightarrow{OA}+ \\overrightarrow{AB} = \\overrightarrow{OB}\\]\\[\\overrightarrow{AB} = \\overrightarrow{OB}-\\overrightarrow{OA}\\]<\/div>\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<p><strong>Position Vector:<\/strong> If P is any point in the space and 0 is the origin, then <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OP}\\]<\/div>\n\n\n\n<p>                           is called the position vector of the point&nbsp; P.<\/p>\n\n\n\n<p>                          Let P be a point in a Three dimensional Space.  Let 0 be the origin and i, j and k&nbsp; the unit vectors along the x ,yand&nbsp; z&nbsp; axes . Then if P is (x, y, z) , the position vector of the point P is <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OP}= x\\overrightarrow{i}\\ + y\\overrightarrow{j}+ z\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[OP =\\overrightarrow{|OP|} = \\sqrt{x^2 + y^2 + z^2 }\\]<\/div>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Direction Cosines<\/strong> <strong>and  Direction ratios<\/strong><\/h4>\n\n\n\n<p>When a directed line&nbsp;<em>OP<\/em>&nbsp;passing through the origin makes&nbsp;\u03b1,&nbsp;\u03b2&nbsp;and&nbsp;\u03b3&nbsp;angles with the&nbsp;x,&nbsp;y and&nbsp;z axis respectively with&nbsp;<em>O<\/em>&nbsp;as the reference, these angles are referred as the direction angles of the line and the cosine of these angles give us the direction cosines. These direction cosines are usually represented as l, m and n.<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter is-resized\"><img data-recalc-dims=\"1\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/cdn1.byjus.com\/wp-content\/uploads\/2018\/10\/0-37.png?ssl=1\" alt=\"Direction Cosines &amp; Direction Ratios - Definitions &amp; Examples\" style=\"width:370px;height:354px\"\/><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OP}= x\\overrightarrow{i}\\ + y\\overrightarrow{j}+ z\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[OP =\\overrightarrow{|OP|} = \\sqrt{x^2 + y^2 + z^2 }= r\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cos\\ \u03b1\\ =\\ \\frac{x}{r},\\ cos\\ \u03b2\\ =\\ \\frac{y}{r},\\ cos\\ \u03b3\\ =\\ \\frac{z}{r}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Direction\\  cosines\\  are \\frac{x}{r},  \\frac{y}{r},    \\frac{z}{r}  \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Result:\\ for\\ any\\ vector\\ the\\ sum\\ of\\ squares\\ of\\ direction\\ cosines\\ of\\ \\overrightarrow{r}\\ is\\ 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ i.e\\ cos^2\\  \u03b1\\ +\\ cos^2\\  \u03b2\\ +\\ cos^2\\  \u03b3\\ =\\ 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Direction\\ ratio\\ are\\  x,\\ y,\\ z\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 2\\ .}\\ \\color {red} {Find\\ the\\  Direction\\  cosines\\ and\\ Direction\\ ratios}\\ of\\ the\\ vector\\  3\\overrightarrow{i}\\ + 4\\overrightarrow{j}- 5\\overrightarrow{k}\\]<\/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}= 3\\overrightarrow{i}\\ + 4\\overrightarrow{j}- 5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[r =\\overrightarrow{|a|} = \\sqrt{(3)^2 + (4)^2 + (-5)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(9 + 16 +25 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[r =\\sqrt{50}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Direction\\  cosines\\  are \\frac{3}{\\sqrt(50)},  \\frac{4}{\\sqrt(50)},    \\frac{-5}{\\sqrt(50)}  \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Direction\\  ratios\\  are\\ 3,   4,   -5  \\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/0M2rGUAW6aU\" title=\"Vector Introduction - Part - 2\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscree=\"\"n><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 3\\ .}\\ \\color {red} {Can\\ a\\ vector\\ have\\ direction\\ angles}\\ 30^0,\\ 45^0,\\ 60^0\\ \\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\">\\[The\\ condition\\ is\\ cos^2\\  \u03b1\\ +\\ cos^2\\  \u03b2\\ +\\ cos^2\\  \u03b3\\ =\\ 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ \u03b1\\ =\\ 30^0,\\ \u03b2\\ =\\ 45^0,\\ \u03b3\\ =\\ 60^0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cos^2\\  \u03b1\\ +\\ cos^2\\  \u03b2\\ +\\ cos^2\\  \u03b3\\ =\\ cos^2\\  30^0\\ +\\ cos^2\\  45^0\\ +\\ cos^2\\  60^0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ (\\frac{\\sqrt{3}}{2})^2\\ +\\ (\\frac{1}{\\sqrt{2}})^2\\ +\\ (\\frac{1}{2})^2\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{3}{4}\\ +\\ \\frac{1}{2}\\ +\\ \\frac{1}{4}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{3\\ +\\ 2\\ +\\ 1}{4}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{6}{4}\\ \\neq 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed {\\therefore\\ they\\ are\\ not\\ directions\\ angles\\ of\\ any\\ vector}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/Qt2fEV2zXHg\" title=\"Vector Introduction - Part - 3\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"aicp\"><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<h4 class=\"wp-block-heading has-vivid-cyan-blue-color has-text-color\"><strong>Distance between two points:<\/strong><\/h4>\n\n\n\n<p>If A and B are two points in the space with co-ordinates A (x<sub>1<\/sub>, y<sub>1<\/sub>, z<sub>1<\/sub> ) and B (x<sub>2<\/sub>, y<sub>2<\/sub>, z<sub>2<\/sub>), then the position vectors are <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}= x_1\\overrightarrow{i}\\ + y_1\\overrightarrow{j}+ z_1\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}= x_2\\overrightarrow{i}\\ + y_2\\overrightarrow{j}+ z_2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB =\\overrightarrow{|AB|} = \\overrightarrow{|OB &#8211; BA|} = \\sqrt{(x_2 &#8211; x_1)^2 + (y_2 &#8211; y_1)^2 +(z_2 &#8211; z_1)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 4\\ .}\\ If\\ position\\ vectors\\ of\\ the\\ points\\ A\\ and\\ B\\ are\\ \\hspace{8cm}\\]\\[\\overrightarrow{i}\\ + 2\\overrightarrow{j}- 3\\overrightarrow{k}\\ and\\  2\\overrightarrow{i}\\ &#8211; \\overrightarrow{j},\\ \\color {red} {find\\ \\overrightarrow{|AB|}}\\ \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ Given\\ \\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/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}= 2\\overrightarrow{i}\\ &#8211; \\overrightarrow{j}\\]<\/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}\\ &#8211; \\overrightarrow{j}- (\\overrightarrow{i}\\ + 2\\overrightarrow{j}- 3\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=2\\overrightarrow{i}\\ &#8211; \\overrightarrow{j}- \\overrightarrow{i}\\ &#8211; 2\\overrightarrow{j} + 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}= \\overrightarrow{i} &#8211; 3\\overrightarrow{j} + 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB =\\overrightarrow{|AB|} = \\sqrt{(1)^2 + (-3)^2 +(3)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(1 + 9 +9 })\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{19}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/lIaX1z9AqRA\" title=\"Vector Introduction - 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<!-- 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\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<p>        Defn:  If  a is a  vector,&nbsp; then <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {Unit\\  vector\\  along\\ \\overrightarrow{a}=\\frac{\\overrightarrow{a}}{\\overrightarrow{|a|}}}\\]<\/div>\n\n\n\n<p><\/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\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 5\\ .}\\ \\color {red} {Find\\ the\\  Unit\\ vector\\ along\\ the\\ vector}\\  3\\overrightarrow{i}\\ + 4\\overrightarrow{j}- 5\\overrightarrow{k}\\ \\hspace{15cm}\\]<\/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}= 3\\overrightarrow{i} + 4\\overrightarrow{j} &#8211; 5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{|a|} = \\sqrt{(3)^2 + (4)^2+(-5)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\sqrt{(9 + 16 +25}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\sqrt{50}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{|a|}=\\sqrt{50}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Unit\\  vector\\  along\\ \\overrightarrow{a}=\\frac{\\overrightarrow{a}}{\\overrightarrow{|a|}}= \\frac{3\\overrightarrow{i}\\ + 4\\overrightarrow{j}- 5\\overrightarrow{k}}{\\sqrt{50}}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/6Ab9EMJvWao\" title=\"Vector Introduction - 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\\ 6\\ .}\\ \\color {red} {Find\\ the\\  Unit\\ vector\\ parallel\\ to\\  2\\overrightarrow{i}\\ &#8211; \\overrightarrow{j}\\ +\\ 4\\overrightarrow{k}}\\ \\hspace{15cm}\\]<\/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}\\ =\\ 2\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ +\\ 4\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{|a|} = \\sqrt{(2)^2 + (-1)^2+(4)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\sqrt{(4\\ +\\ 1\\ +\\ 16)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\sqrt{21}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{|a|}=\\sqrt{21}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Unit\\  vector\\  along\\ \\overrightarrow{a}=\\frac{\\overrightarrow{a}}{\\overrightarrow{|a|}}= \\frac{3\\overrightarrow{i}\\ + 4\\overrightarrow{j}- 5\\overrightarrow{k}}{\\sqrt{50}}\\]<\/div>\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:inline-block;width:300px;height:250px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"5990643685\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\underline{\\color {red} {Condition\\ for\\ two\\ position\\ vectors\\ \\overrightarrow{a}\\ and\\  \\overrightarrow{b}\\ to\\ be\\ collinear}}\\ if\\ \\color {green} {\\overrightarrow{a}\\ =\\ k\\ \\overrightarrow{b}}\\ \\hspace{10cm}\\] <\/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\">\\[\\underline{\\color {red} {Condition\\ for\\ three\\ position\\ vectors\\ \\overrightarrow{OA}\\ ,\\ \\overrightarrow{OB}\\ and\\  \\overrightarrow{OC}\\ to\\ be}}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[i)\\  \\color {green} {collinear \\  if  \\overrightarrow{BC} = K\\overrightarrow{AB}} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ii) \\color {green} {Equilateral\\ triangle\\ if\\  \\overrightarrow{|AB|}=\\overrightarrow{|BC|}=\\overrightarrow{|AC|}} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[iii) \\color {green} {Isosceles\\ triangle\\ if\\  \\overrightarrow{|AB|}=\\overrightarrow{|BC|}\\not=\\overrightarrow{|AC|}  or\\ \\overrightarrow{|BC|}=\\overrightarrow{|AC|}\\not=\\overrightarrow{|AB|}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[iv) \\color {green} {right\\ angled\\ triangle\\ if\\ (AB)^2+(BC)^2=(AC)^2  or\\   (BC)^2+(AC)^2=(AB)^2} \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 7\\ .}\\ If\\ the\\ vectors\\ 2\\overrightarrow{i}\\ -\\  3\\overrightarrow{j}\\ and\\ 6\\overrightarrow{i}\\ -\\  m\\overrightarrow{j}\\ are\\ collinear,\\ \\hspace{10cm}\\]\\[\\color {red} {find\\ the\\ value\\ of\\ m}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{19cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\\ =\\ 2\\overrightarrow{i}\\ &#8211; 3\\overrightarrow{j}\\ and\\ \\overrightarrow{b}\\ =\\ 6\\overrightarrow{i}\\ -\\ m\\overrightarrow{j}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ that\\ \\overrightarrow{a}\\ and\\ \\overrightarrow{b}\\ are\\ collinear\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}\\ =\\ k\\ \\overrightarrow{b}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\overrightarrow{i}\\ &#8211; 3\\overrightarrow{j}\\ =\\ k\\ (6\\overrightarrow{i}\\ -\\ m\\overrightarrow{j})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Comparing\\ LHS\\ and\\ RHS\\ we\\ get\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ =\\ 6k\\ &#8212; (1)\\ \\hspace{2cm}\\ and\\ \\hspace{2cm}\\ -\\ 3\\ =\\ -\\ mk\\ &#8212; (2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[k\\ =\\ \\frac{1}{3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Put\\ k\\ value\\ in\\ (2)\\ we\\ get\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[-\\ 3\\ =\\ -\\ m (\\frac{1}{3})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[m\\ =\\ 9\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/sWwUuCCFTnU\" title=\"Vector Introduction - Part - 6\" 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\">\\[\\color {purple} {Example\\ 8\\ .}\\  \\color {red} {Show\\ that\\ the\\ points\\ whose\\ position\\ vectors}\\ \\hspace{12cm}\\]\\[2\\overrightarrow{i}\\ +\\  3\\overrightarrow{j}\\ -\\  5\\overrightarrow{k},\\  3\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -\\  2\\overrightarrow{k}\\ and\\ 6\\overrightarrow{i}\\ -\\ 5 \\overrightarrow{j}\\ +\\ 7\\overrightarrow{k}\\ \\color {red} {are\\ collinear}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ Given\\ \\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}= 2\\overrightarrow{i}\\ + 3\\overrightarrow{j}- 5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}= 3\\overrightarrow{i}\\ +\\overrightarrow{j} &#8211; 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OC}= 6\\overrightarrow{i}\\ &#8211; 5\\overrightarrow{j} + 7\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB} = \\overrightarrow{OB}-\\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=3\\overrightarrow{i}+ \\overrightarrow{j} &#8211; 2\\overrightarrow{k}- (2\\overrightarrow{i}\\ + 3\\overrightarrow{j}- 5\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=3\\overrightarrow{i}+ \\overrightarrow{j} &#8211; 2\\overrightarrow{k}- 2\\overrightarrow{i}\\ &#8211; 3\\overrightarrow{j}+ 5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}=  \\overrightarrow{i} &#8211; 2\\overrightarrow{j} + 3\\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\">\\[=6\\overrightarrow{i}- 5 \\overrightarrow{j} + 7\\overrightarrow{k}- (3\\overrightarrow{i}\\ + \\overrightarrow{j}- 2\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=6\\overrightarrow{i}-5 \\overrightarrow{j} +7\\overrightarrow{k}- 3\\overrightarrow{i}\\ &#8211; \\overrightarrow{j}+ 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{BC}=  3\\overrightarrow{i} &#8211; 6\\overrightarrow{j} + 9\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 3( \\overrightarrow{i}\\ &#8211; 2\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{BC}\\ =\\ 3\\ \\overrightarrow{AB}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA},\\overrightarrow{OB}\\ and\\  \\overrightarrow{OC} \\  are\\  collinear\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/ygB07COf05M\" title=\"Vector Introduction - 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=\"aicp\"><script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block; text-align:center;\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"2812384453\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/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\\ 9\\ .}\\  \\color {red} {Prove\\ that\\ the\\ points}\\ \\hspace{15cm}\\]\\[4\\overrightarrow{i}\\ + 2\\overrightarrow{j}+ 3\\overrightarrow{k},  2\\overrightarrow{i}\\ + 3\\overrightarrow{j}+ 4\\overrightarrow{k} and\\ 3\\overrightarrow{i}\\ +4 \\overrightarrow{j}+ 2\\overrightarrow{k}\\  \\color {red} {form\\  an\\ equilateral\\  triangle}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ Given\\ \\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}= 4\\overrightarrow{i}\\ + 2\\overrightarrow{j}+ 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}= 2\\overrightarrow{i}\\ +3\\overrightarrow{j} + 4\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OC}= 3\\overrightarrow{i}\\ + 4\\overrightarrow{j} + 2\\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} + 4\\overrightarrow{k}- (4\\overrightarrow{i}+ 2\\overrightarrow{j}+ 3\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=2\\overrightarrow{i}+ 3\\overrightarrow{j} + 4\\overrightarrow{k}- 4\\overrightarrow{i}- 2\\overrightarrow{j}- 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}=  -2\\overrightarrow{i} +\\overrightarrow{j} + \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB =\\overrightarrow{|AB|} = \\sqrt{(-2)^2 + (-1)^2 +(1)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(4 + 1 +1 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB = \\sqrt{6}\\]<\/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\">\\[=3\\overrightarrow{i}+ 4\\overrightarrow{j} + 2\\overrightarrow{k}- (2\\overrightarrow{i}+ 3\\overrightarrow{j}+ 4\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=3\\overrightarrow{i}+ 4\\overrightarrow{j} + 2\\overrightarrow{k}- 2\\overrightarrow{i}- 3\\overrightarrow{j}- 4\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{BC}=  \\overrightarrow{i} +\\overrightarrow{j} -2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[BC =\\overrightarrow{|BC|} = \\sqrt{(1)^2 + (1)^2 +(-2)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(1 + 1 + 4}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[BC = \\sqrt{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AC} = \\overrightarrow{OC}-\\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=3\\overrightarrow{i}+ 4\\overrightarrow{j} + 2\\overrightarrow{k}- (4\\overrightarrow{i}+ 2\\overrightarrow{j}+ 3\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=3\\overrightarrow{i}+ 4\\overrightarrow{j} + 2\\overrightarrow{k}- 4\\overrightarrow{i}- 2\\overrightarrow{j}- 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AC}=  -\\overrightarrow{i} +2\\overrightarrow{j} -\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AC =\\overrightarrow{|AC|} = \\sqrt{(-1)^2 + (2)^2 +(-1)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(1 + 4 + 1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AC = \\sqrt{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB = BC = AC = \\sqrt{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<p>The given triangle is an equilateral triangle.<\/p>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/OsM0lZ2G2FA\" title=\"Vector Introduction - 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\\ 10\\ .}\\  \\color {red} {Prove\\ that\\ the\\ points}\\ \\hspace{15cm}\\]\\[2\\overrightarrow{j}\\  +\\ 10 \\overrightarrow{k}\\ ,\\  7\\overrightarrow{i}\\ +\\  6\\overrightarrow{j}\\ +\\  6\\overrightarrow{k} and\\ -\\ 4\\overrightarrow{i}\\  +\\ 9 \\overrightarrow{j}\\  +\\ 6 \\overrightarrow{k}\\  \\color {red} {form\\  an\\ isosceles\\ triangle}\\]<\/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\\ :}\\ Given\\ \\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}\\ =\\ 2\\ \\overrightarrow{j}\\  +\\ 10 \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}\\ =\\ 7\\overrightarrow{i}\\ +\\  6\\overrightarrow{j}\\ +\\  6\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OC}\\ =\\ -\\ 4\\overrightarrow{i}\\  +\\ 9 \\overrightarrow{j}\\  +\\ 6 \\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\">\\[=\\ 7\\overrightarrow{i}\\ +\\  6\\overrightarrow{j}\\ +\\  6\\overrightarrow{k}\\ &#8211; (2\\ \\overrightarrow{j}\\  +\\ 10 \\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 7\\overrightarrow{i}\\ +\\  6\\overrightarrow{j}\\ +\\  6\\overrightarrow{k}\\ -\\  2 \\overrightarrow{j}\\ -\\  10\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}\\ =\\  7\\overrightarrow{i}\\ +\\  4\\overrightarrow{j}\\ -\\ 4\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB =\\overrightarrow{|AB|} = \\sqrt{(7)^2\\ +\\  (4)^2\\ +\\ (-4)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(49\\ +\\ 16\\ +\\ 16)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\  \\sqrt{81}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB =\\ 9 \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{BC} = \\overrightarrow{OC}-\\overrightarrow{OB}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ -\\ 4\\overrightarrow{i}\\  +\\ 9 \\overrightarrow{j}\\  +\\ 6 \\overrightarrow{k}\\ &#8211; (7\\overrightarrow{i}\\ +\\  6\\overrightarrow{j}\\ +\\  6\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ -\\ 4\\overrightarrow{i}\\  +\\ 9 \\overrightarrow{j}\\  +\\ 6 \\overrightarrow{k}\\ -\\  7\\overrightarrow{i}\\ &#8211; 6\\ \\overrightarrow{j}\\ -\\  6\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{BC}\\ =\\  -\\ 11\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[BC =\\overrightarrow{|BC|} = \\sqrt{(-11)^2 + (3)^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{121\\ +\\ 9}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\  \\sqrt{130}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AC} = \\overrightarrow{OC}-\\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ -\\ 4\\overrightarrow{i}\\  +\\ 9 \\overrightarrow{j}\\  +\\ 6 \\overrightarrow{k}\\ &#8211; (2\\ \\overrightarrow{j}\\  +\\ 10 \\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ -\\ 4\\overrightarrow{i}\\  +\\ 9 \\overrightarrow{j}\\  +\\ 6 \\overrightarrow{k}\\ -\\ 2\\ \\overrightarrow{j}\\ -\\  10\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AC}\\ =\\  -\\ 4\\overrightarrow{i}\\ +\\  7\\overrightarrow{j}\\ -\\ 4\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AC =\\overrightarrow{|AC|} = \\sqrt{(-\\ 4)^2\\ +\\  (7)^2\\ +\\ (-\\ 4)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{16\\ +\\  49\\ +\\  16}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AC = \\sqrt{81}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{AB = AC\\ \\neq\\ BC}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ given\\ triangle\\ is\\ isosceles\\ triangle\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/HO_9-tRDDjM\" title=\"Vector Introduction - Part - 9\" 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\\ 11\\ .}\\  \\color {red} {Prove\\ that\\ the\\ points\\ whose\\ position\\ vectors\\ are}\\ \\hspace{15cm}\\]\\[3\\overrightarrow{i}\\ &#8211; \\overrightarrow{j}+ 6\\overrightarrow{k},  5\\overrightarrow{i}\\ &#8211; 2\\overrightarrow{j}+ 7\\overrightarrow{k} and\\ 6\\overrightarrow{i}\\ -5 \\overrightarrow{j}+ 2\\overrightarrow{k}\\ \\color {red} {form\\  a\\ right\\   angled\\ triangle}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ Given\\ \\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}= 3\\overrightarrow{i} &#8211; \\overrightarrow{j}+ 6\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}= 5\\overrightarrow{i}\\ -2\\overrightarrow{j} + 7\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OC}= 6\\overrightarrow{i}\\ &#8211; 5\\overrightarrow{j} + 2\\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\">\\[=5\\overrightarrow{i}- 2\\overrightarrow{j} + 7\\overrightarrow{k}- (3\\overrightarrow{i}- \\overrightarrow{j}+ 6\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=5\\overrightarrow{i}- 2\\overrightarrow{j} + 7\\overrightarrow{k}- 3\\overrightarrow{i}+ \\overrightarrow{j}- 6\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}=  2\\overrightarrow{i} -\\overrightarrow{j} + \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB =\\overrightarrow{|AB|} = \\sqrt{(2)^2 + (-1)^2 +(1)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(4 + 1 +1 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB = \\sqrt{6}\\]<\/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\">\\[=6\\overrightarrow{i}- 5\\overrightarrow{j} + 2\\overrightarrow{k}- (5\\overrightarrow{i}- 2\\overrightarrow{j}+ 7\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=6\\overrightarrow{i}- 5\\overrightarrow{j} + 2\\overrightarrow{k}- 5\\overrightarrow{i}+ 2\\overrightarrow{j}- 7\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{BC}=  \\overrightarrow{i} -3\\overrightarrow{j} -5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[BC =\\overrightarrow{|BC|} = \\sqrt{(1)^2 + (-3)^2 +(-5)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(1 + 9 + 25}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[BC = \\sqrt{35}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AC} = \\overrightarrow{OC}-\\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=6\\overrightarrow{i}- 5\\overrightarrow{j} + 2\\overrightarrow{k}- (3\\overrightarrow{i}- \\overrightarrow{j}+ 6\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=6\\overrightarrow{i}- 5\\overrightarrow{j} + 2\\overrightarrow{k}- 3\\overrightarrow{i}+\\overrightarrow{j}- 6\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AC}= 3\\overrightarrow{i} -4\\overrightarrow{j} -4\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AC =\\overrightarrow{|AC|} = \\sqrt{(3)^2 + (-4)^2 +(-4)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(9 + 16 + 16}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AC = \\sqrt{41}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB^2 = 6,    BC^2 = 35,          AC^2 =  41 \\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB^2 + BC^2 = 6 + 35 = 41=AC^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB^2 + BC^2 = AC^2\\]<\/div>\n\n\n\n<p>The given triangle is an right angled triangle.<\/p>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/WUXxqqeAj5U\" title=\"Vector Introduction - 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=\"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\">\\[ Let\\ \\overrightarrow{a}\\ =\\ a_1\\ \\overrightarrow{i}\\ +\\ a_2\\ \\overrightarrow{j}\\ +\\ a_3\\ \\overrightarrow{k}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}\\ =\\ b_1\\ \\overrightarrow{i}\\ +\\ b_2\\ \\overrightarrow{j}\\ +\\ b_3\\ \\overrightarrow{k}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{c}\\ =\\ c_1\\ \\overrightarrow{i}\\ +\\ c_2\\ \\overrightarrow{j}\\ +\\ c_3\\ \\overrightarrow{k}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Then\\ [\\overrightarrow{a}\\ \\overrightarrow{b}\\ \\overrightarrow{c}]\\ =\\begin{vmatrix}\na_1 &amp;amp: a_2 &amp;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} {Condition\\ for\\ three\\ vectors\\ \\overrightarrow{a}\\  \\overrightarrow{b}\\ and\\  \\overrightarrow{c}\\ to\\ be\\ coplanar}}\\ if\\ \\color {green} {[\\overrightarrow{a}\\ \\overrightarrow{b}\\ \\overrightarrow{c}]\\ =\\ 0}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\underline{\\color {red} {Condition\\ for\\ three\\ position\\ vectors\\ \\overrightarrow{OA}\\ ,\\ \\overrightarrow{OB}\\ \\overrightarrow{OC}\\ and\\ \\overrightarrow{OD}\\ to\\ be\\ coplanar}}\\ if\\ \\color {green} {[\\overrightarrow{AB}\\   \\overrightarrow{AC}\\  \\overrightarrow{AD}]\\ =\\ 0}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 12\\ .}\\  \\color {red} {Find\\ the\\ value\\ of\\ p}\\ such\\ that\\ the\\ vectors\\ \\hspace{15cm}\\]\\[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}\\ \\color {red} {lie\\  on\\ the\\   same\\ plane}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ Given\\ \\hspace{17cm}\\]<\/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}\\ -\\  6\\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\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[[\\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; -6 \\\\\n3 &amp; -1 &amp; 4 \\\\\n\\end{vmatrix}\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\begin{vmatrix}\n2 &amp;  -6 \\\\\n-1 &amp;  4 \\\\\n\\end{vmatrix}\\ +\\ 3\\begin{vmatrix}\np &amp;  -6 \\\\\n3 &amp; 4\\\\\n\\end{vmatrix}\\  +\\ 5\\begin{vmatrix}\np &amp; 2 \\\\\n3 &amp; &#8211; 1\\\\\n\\end{vmatrix}\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2(8\\ -\\ 6)\\ +\\ 3 (4p\\ +\\ 18)\\ +\\ 5(-p\\ -\\ 6)\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ 4\\ +\\ 12\\ p\\ +\\ 54\\ -\\ 5\\ p\\ -\\ 30\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[7p\\ +\\ 58\\  -\\  30\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[7p\\ +\\ 28\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[7p\\  =\\ -\\ 28\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{p\\ =\\ -\\ 4}\\]   <\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/tZn5NJyzzlk\" title=\"Vector Introduction - Part - 11\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 13\\ .}\\  \\color {red} {Prove\\ that\\ the\\ points\\ given\\ by\\ the\\ position\\ vectors}\\  \\hspace{15cm}\\]\\[4\\overrightarrow{i}\\ +\\ 5\\ \\overrightarrow{j}\\ +\\  \\overrightarrow{k},  -\\ \\overrightarrow{j}\\ -\\ \\overrightarrow{k},  3\\overrightarrow{i}\\ +\\ 9\\overrightarrow{j}\\ +\\  4\\overrightarrow{k}\\ and\\ -4\\ \\overrightarrow{i}\\ +\\ 4\\overrightarrow{j}\\ +\\  4\\overrightarrow{k}\\ \\color {red} {are\\ coplanar}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln\\ :}\\ Given\\ \\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}\\ =\\ 4\\overrightarrow{i}\\ +\\ 5\\ \\overrightarrow{j}\\ +\\  \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}\\ =\\ -\\ \\overrightarrow{j}\\ -\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OC}\\ =\\ 3\\overrightarrow{i}\\ +\\ 9\\overrightarrow{j}\\ +\\  4\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OD}\\ =\\ -4\\ \\overrightarrow{i}\\ +\\ 4\\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{j}\\ -\\ \\overrightarrow{k}\\ &#8211; (4\\overrightarrow{i}\\ +\\ 5\\ \\overrightarrow{j}\\ +\\  \\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ -\\ \\overrightarrow{j}\\ -\\  \\overrightarrow{k}\\ -\\ 4\\overrightarrow{i}\\ -\\ 5\\overrightarrow{j}\\ -\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}\\ =\\  -\\ 4\\overrightarrow{i}\\ -\\ 6\\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\">\\[=\\ 3\\overrightarrow{i}\\ +\\ 9\\overrightarrow{j}\\ +\\  4\\overrightarrow{k}\\ -\\ (-\\ \\overrightarrow{j}\\ -\\ \\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 3\\overrightarrow{i}\\ +\\ 9\\overrightarrow{j}\\ +\\ 4\\overrightarrow{k}\\ +\\ \\overrightarrow{j}\\ +\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{BC}\\ =\\ 3\\  \\overrightarrow{i}\\  +\\ \\ 10\\overrightarrow{j}\\  +\\ 5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AC} = \\overrightarrow{OC}-\\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 3\\overrightarrow{i}\\ +\\ 9\\overrightarrow{j}\\ +\\  4\\overrightarrow{k}\\ -\\ (4\\overrightarrow{i}\\ +\\ 5\\ \\overrightarrow{j}\\ +\\  \\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 3\\overrightarrow{i}\\ +\\ 9\\overrightarrow{j}\\ +\\  4\\overrightarrow{k}\\ -\\ 4\\overrightarrow{i}\\ -\\ 5\\overrightarrow{j}\\ -\\  \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AC}\\ =\\  -\\ \\overrightarrow{i}\\ +\\ 4\\ \\overrightarrow{j}\\ +\\ 3\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AD} = \\overrightarrow{OD}-\\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ -4\\ \\overrightarrow{i}\\ +\\ 4\\overrightarrow{j}\\ +\\  4\\overrightarrow{k}\\ -\\ (4\\overrightarrow{i}\\ +\\ 5\\ \\overrightarrow{j}\\ +\\  \\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ -4\\ \\overrightarrow{i}\\ +\\ 4\\overrightarrow{j}\\ +\\  4\\overrightarrow{k}\\ -\\ 4\\overrightarrow{i}\\ -\\ 5\\overrightarrow{j}\\ -\\  \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AD}\\ =\\  -\\ 8\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ +\\ 3\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[To\\ claim\\ \\overrightarrow{AB}\\  \\overrightarrow{AC}\\ and\\  \\overrightarrow{AD}\\ coplanar\\ need\\ to\\ prove\\ [\\overrightarrow{AB}\\ \\overrightarrow{AC}\\ \\overrightarrow{AD}]\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[[\\overrightarrow{AB}\\ \\overrightarrow{AC}\\ \\overrightarrow{AD}]\\ =\\ \\begin{vmatrix}\n&#8211; 4 &amp; &#8211; 6 &amp; &#8211; 2 \\\\\n-1 &amp; 4 &amp; 3 \\\\\n-8 &amp; -1 &amp; 3 \\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ -4\\begin{vmatrix}\n4 &amp;  3 \\\\\n-1 &amp;  3 \\\\\n\\end{vmatrix}\\ +\\ 6\\begin{vmatrix}\n-1 &amp;  3 \\\\\n-8 &amp; 3\\\\\n\\end{vmatrix}\\  -\\ 2\\begin{vmatrix}\n-1 &amp; 4 \\\\\n-8 &amp; &#8211; 1\\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ -4\\ (12\\ +\\ 3)\\ +\\ 6 (-3\\ +\\ 24)\\ -\\ 2(1\\ +\\ 32)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ -\\ 4\\ (15)\\ +\\ 6\\ (21)\\ -\\ 2\\ (33)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ -\\ 60\\ +\\ 126\\  -\\  66\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ -\\ 126\\ +\\ 126\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ the\\ given\\ position\\ vectors\\  \\overrightarrow{AB}\\  \\overrightarrow{AC}\\ and\\  \\overrightarrow{AD}\\  are\\ coplanar\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/jdG6hH18gHE\" title=\"Vector Introduction - Part - 12\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-vivid-cyan-blue-color has-text-color\">Exercise Problems<\/h3>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- A}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {1\\ .}\\ \\overrightarrow{a}= 3\\overrightarrow{i}\\ + 2\\overrightarrow{j} + \\overrightarrow{k}\\ and\\  \\overrightarrow{b}= \\overrightarrow{i}\\ + 3\\overrightarrow{j} + \\overrightarrow{k},\\ \\color {red} {find\\   3\\overrightarrow{a}\\ +\\   \\overrightarrow{b}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {2\\ .}\\ \\overrightarrow{a}= 2\\overrightarrow{i}\\ + 3\\overrightarrow{j} + \\overrightarrow{k}\\ and\\  \\overrightarrow{b}= 3\\overrightarrow{i}\\ &#8211; \\overrightarrow{j} + \\overrightarrow{k},\\ \\color {red} {find\\   2\\overrightarrow{a}\\ +\\   3\\overrightarrow{b}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {3\\ .}\\ If\\ position\\ vectors\\ of\\ the\\ points\\ A\\ and\\ B\\ are\\  2\\overrightarrow{i}\\ -\\overrightarrow{j} +  3\\overrightarrow{k}\\ and\\ 5\\overrightarrow{i}\\ + \\overrightarrow{j} &#8211; 2\\overrightarrow{k},\\ \\color {red} {find\\  \\overrightarrow{|AB|}}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {4\\ .}\\ \\color {red} {Find\\ the\\ unit\\ vector\\ along\\ the\\ vector}\\ 2\\overrightarrow{i}\\ &#8211; \\overrightarrow{j}- \\overrightarrow{k}\\ \\hspace{20cm}\\]<\/div>\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\">\\[\\LARGE{\\color {purple} {PART- B}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {5\\ .}\\ If\\ the\\ position\\ vector\\ of\\ the\\ points\\ A\\ and\\ B\\ are\\ \\hspace{15cm}\\]\\[\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ +\\  \\overrightarrow{k}\\ and\\  3\\overrightarrow{i}\\ +\\  2\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k},\\ \\hspace{12cm}\\]\\[\\color {red} {find\\ \\overrightarrow{|AB|}\\ ,\\ Also\\ find\\ the\\ direction\\ ratio\\ of\\ \\overrightarrow{AB}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {6\\ .}\\ \\color {red} {Find\\ the\\  Modulus\\ and\\ Direction\\  cosines}\\ \\hspace{15cm}\\]\\[of\\ the\\ vector\\  2\\overrightarrow{i}\\ + 3\\overrightarrow{j}\\ +\\ 4\\overrightarrow{k}\\ \\hspace{12cm}\\]<\/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\">\\[\\LARGE{\\color {purple} {PART- C}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {7\\ .}\\ \\color {red} {\\  Show\\ that}\\ the\\ points\\ whose\\ position\\ vectors\\ \\hspace{15cm}\\]\\[\\overrightarrow{i}\\ -\\  2\\overrightarrow{j}\\ -\\  \\overrightarrow{k},\\  2\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\  3\\overrightarrow{k}\\ and\\  -\\ 7 \\overrightarrow{j}\\ -\\ 5\\overrightarrow{k}\\ \\color {red} {are\\ collinear}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {8\\ .}\\ \\color {red} {Prove\\ that\\ the\\ points}\\ \\hspace{15cm}\\]\\[2\\overrightarrow{i}\\ + 3\\overrightarrow{j}+ 4\\overrightarrow{k},  3\\overrightarrow{i}\\ + 4\\overrightarrow{j}\\ + 2\\overrightarrow{k} and\\ 4\\overrightarrow{i}\\ +\\ 2 \\overrightarrow{j}+ 3\\overrightarrow{k}\\  \\color {red} {form\\  an\\ equilateral\\  triangle}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {9\\ .}\\ \\color {red} { Prove\\ that\\ the\\ points}\\ \\hspace{15cm}\\]\\[3\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\  -\\ 2 \\overrightarrow{k}\\ ,\\  5\\overrightarrow{i}\\ +\\  \\overrightarrow{j}\\ -\\  3\\overrightarrow{k} and\\ 6\\overrightarrow{i}\\  -\\  \\overrightarrow{j}\\  -\\  \\overrightarrow{k}\\  \\color {red} {form\\  an\\ isosceles\\ triangle}\\]<\/div>\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\"><\/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<p><\/p>\n\n\n\n<a target=\"_blank\" href=\"https:\/\/www.amazon.in\/gp\/product\/B08CXQWT9Z\/ref=as_li_tl?ie=UTF8&amp;camp=3638&amp;creative=24630&amp;creativeASIN=B08CXQWT9Z&amp;linkCode=as2&amp;tag=yanamtakshash-21&amp;linkId=ba081edb3c707ccf0ec27b9f15c521e4\" rel=\"noopener\"><img decoding=\"async\" border=\"0\" src=\"\/\/ws-in.amazon-adsystem.com\/widgets\/q?_encoding=UTF8&amp;MarketPlace=IN&amp;ASIN=B08CXQWT9Z&amp;ServiceVersion=20070822&amp;ID=AsinImage&amp;WS=1&amp;Format=_SL160_&amp;tag=yanamtakshash-21\"><\/a>\n\n\n\n<a target=\"_blank\" href=\"https:\/\/www.amazon.in\/gp\/product\/B085HGZVN8\/ref=as_li_tl?ie=UTF8&amp;camp=3638&amp;creative=24630&amp;creativeASIN=B085HGZVN8&amp;linkCode=as2&amp;tag=yanamtakshash-21&amp;linkId=8515ec5d06cd7d5a1d0f1408b2a416a7\" rel=\"noopener\"><img decoding=\"async\" border=\"0\" src=\"\/\/ws-in.amazon-adsystem.com\/widgets\/q?_encoding=UTF8&amp;MarketPlace=IN&amp;ASIN=B085HGZVN8&amp;ServiceVersion=20070822&amp;ID=AsinImage&amp;WS=1&amp;Format=_SL160_&amp;tag=yanamtakshash-21\"><\/a>\n\n\n\n<a target=\"_blank\" href=\"https:\/\/www.amazon.in\/gp\/product\/B08JLKSW5N\/ref=as_li_tl?ie=UTF8&amp;camp=3638&amp;creative=24630&amp;creativeASIN=B08JLKSW5N&amp;linkCode=as2&amp;tag=yanamtakshash-21&amp;linkId=d0163d16872fa6dd94454f84b3afe7e9\" rel=\"noopener\"><img decoding=\"async\" border=\"0\" src=\"\/\/ws-in.amazon-adsystem.com\/widgets\/q?_encoding=UTF8&amp;MarketPlace=IN&amp;ASIN=B08JLKSW5N&amp;ServiceVersion=20070822&amp;ID=AsinImage&amp;WS=1&amp;Format=_SL160_&amp;tag=yanamtakshash-21\"><\/a>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Vectors constitute one of the several Mathematical systems which can be usefully employed to provide mathematical handling for certain types of problems in Geometry, Mechanics and other branches of Applied Mathematics. Vectors facilitate mathematical study of such physical quantities as possess Direction in addition to Magnitude. Velocity of a particle, for example, is one such [&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":[711787867,711787868],"tags":[],"class_list":["post-12376","post","type-post","status-publish","format-standard","hentry","category-n-em-ii-unit-ii","category-vector-introduction"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>2.1 VECTOR \u2013 INTRODUCTION - 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=12376\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"2.1 VECTOR \u2013 INTRODUCTION - YANAMTAKSHASHILA\" \/>\n<meta property=\"og:description\" content=\"Vectors constitute one of the several Mathematical systems which can be usefully employed to provide mathematical handling for certain types of problems in Geometry, Mechanics and other branches of Applied Mathematics. Vectors facilitate mathematical study of such physical quantities as possess Direction in addition to Magnitude. Velocity of a particle, for example, is one such [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/yanamtakshashila.com\/?p=12376\" \/>\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=\"2021-02-27T10:12:35+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-04-16T10:42:10+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.studypage.in\/images\/maths\/algebra\/triangle-law-vector-add.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=\"7 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12376#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12376\"},\"author\":{\"name\":\"rajuviswa\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\"},\"headline\":\"2.1 VECTOR \u2013 INTRODUCTION\",\"datePublished\":\"2021-02-27T10:12:35+00:00\",\"dateModified\":\"2024-04-16T10:42:10+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12376\"},\"wordCount\":3040,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\"},\"image\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12376#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/www.studypage.in\\\/images\\\/maths\\\/algebra\\\/triangle-law-vector-add.png\",\"articleSection\":[\"N-EM-II-Unit-II\",\"Vector-Introduction\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12376#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12376\",\"url\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12376\",\"name\":\"2.1 VECTOR \u2013 INTRODUCTION - YANAMTAKSHASHILA\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12376#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12376#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/www.studypage.in\\\/images\\\/maths\\\/algebra\\\/triangle-law-vector-add.png\",\"datePublished\":\"2021-02-27T10:12:35+00:00\",\"dateModified\":\"2024-04-16T10:42:10+00:00\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12376#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12376\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12376#primaryimage\",\"url\":\"https:\\\/\\\/www.studypage.in\\\/images\\\/maths\\\/algebra\\\/triangle-law-vector-add.png\",\"contentUrl\":\"https:\\\/\\\/www.studypage.in\\\/images\\\/maths\\\/algebra\\\/triangle-law-vector-add.png\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=12376#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/yanamtakshashila.com\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"2.1 VECTOR \u2013 INTRODUCTION\"}]},{\"@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":"2.1 VECTOR \u2013 INTRODUCTION - 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=12376","og_locale":"en_US","og_type":"article","og_title":"2.1 VECTOR \u2013 INTRODUCTION - YANAMTAKSHASHILA","og_description":"Vectors constitute one of the several Mathematical systems which can be usefully employed to provide mathematical handling for certain types of problems in Geometry, Mechanics and other branches of Applied Mathematics. Vectors facilitate mathematical study of such physical quantities as possess Direction in addition to Magnitude. Velocity of a particle, for example, is one such [&hellip;]","og_url":"https:\/\/yanamtakshashila.com\/?p=12376","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":"2021-02-27T10:12:35+00:00","article_modified_time":"2024-04-16T10:42:10+00:00","og_image":[{"url":"https:\/\/www.studypage.in\/images\/maths\/algebra\/triangle-law-vector-add.png","type":"","width":"","height":""}],"author":"rajuviswa","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rajuviswa","Est. reading time":"7 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/yanamtakshashila.com\/?p=12376#article","isPartOf":{"@id":"https:\/\/yanamtakshashila.com\/?p=12376"},"author":{"name":"rajuviswa","@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e"},"headline":"2.1 VECTOR \u2013 INTRODUCTION","datePublished":"2021-02-27T10:12:35+00:00","dateModified":"2024-04-16T10:42:10+00:00","mainEntityOfPage":{"@id":"https:\/\/yanamtakshashila.com\/?p=12376"},"wordCount":3040,"commentCount":0,"publisher":{"@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e"},"image":{"@id":"https:\/\/yanamtakshashila.com\/?p=12376#primaryimage"},"thumbnailUrl":"https:\/\/www.studypage.in\/images\/maths\/algebra\/triangle-law-vector-add.png","articleSection":["N-EM-II-Unit-II","Vector-Introduction"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/yanamtakshashila.com\/?p=12376#respond"]}]},{"@type":"WebPage","@id":"https:\/\/yanamtakshashila.com\/?p=12376","url":"https:\/\/yanamtakshashila.com\/?p=12376","name":"2.1 VECTOR \u2013 INTRODUCTION - YANAMTAKSHASHILA","isPartOf":{"@id":"https:\/\/yanamtakshashila.com\/#website"},"primaryImageOfPage":{"@id":"https:\/\/yanamtakshashila.com\/?p=12376#primaryimage"},"image":{"@id":"https:\/\/yanamtakshashila.com\/?p=12376#primaryimage"},"thumbnailUrl":"https:\/\/www.studypage.in\/images\/maths\/algebra\/triangle-law-vector-add.png","datePublished":"2021-02-27T10:12:35+00:00","dateModified":"2024-04-16T10:42:10+00:00","breadcrumb":{"@id":"https:\/\/yanamtakshashila.com\/?p=12376#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/yanamtakshashila.com\/?p=12376"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/yanamtakshashila.com\/?p=12376#primaryimage","url":"https:\/\/www.studypage.in\/images\/maths\/algebra\/triangle-law-vector-add.png","contentUrl":"https:\/\/www.studypage.in\/images\/maths\/algebra\/triangle-law-vector-add.png"},{"@type":"BreadcrumbList","@id":"https:\/\/yanamtakshashila.com\/?p=12376#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/yanamtakshashila.com\/"},{"@type":"ListItem","position":2,"name":"2.1 VECTOR \u2013 INTRODUCTION"}]},{"@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-3dC","_links":{"self":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/12376","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=12376"}],"version-history":[{"count":99,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/12376\/revisions"}],"predecessor-version":[{"id":45716,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/12376\/revisions\/45716"}],"wp:attachment":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=12376"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=12376"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=12376"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}