{"id":14838,"date":"2021-05-12T20:59:41","date_gmt":"2021-05-12T15:29:41","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=14838"},"modified":"2023-08-15T19:53:44","modified_gmt":"2023-08-15T14:23:44","slug":"n-2-1-vector-introduction-exercise-problems-with-solutions","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=14838","title":{"rendered":"N &#8211; 2.1 &#8211; Vector Introduction &#8211; Exercise Problems with Solutions"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- A}}\\]<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {1\\ .}\\ \\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 {blue} {Soln:}\\ Given\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{a}= 3\\overrightarrow{i}\\ + 2\\overrightarrow{j} + \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{b}\\ =\\ \\overrightarrow{i}\\ + 3\\overrightarrow{j} + \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3\\overrightarrow{a}\\ +\\  \\overrightarrow{b}\\ =\\ 3(3\\overrightarrow{i}\\ + 2\\overrightarrow{j} + \\overrightarrow{k})\\ +\\  \\overrightarrow{i}\\ + 3\\overrightarrow{j} + \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 9\\overrightarrow{i}\\ +\\ 6\\overrightarrow{j}\\ +\\  3\\overrightarrow{k}\\ +\\   \\overrightarrow{i}\\ +\\  3\\overrightarrow{j} + \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3\\overrightarrow{a}\\ +\\  \\overrightarrow{b}\\ =\\ 10\\overrightarrow{i}\\ +\\  9\\overrightarrow{j}\\ +\\ 4\\overrightarrow{k}\\ \\hspace{5cm}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/Au4F5xmJYJU\" title=\"Vector Introduction (Exercise) - 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} {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 {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}= 3\\overrightarrow{i}\\ &#8211; \\overrightarrow{j} + \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\overrightarrow{a}\\ +\\   3\\overrightarrow{b}= 2(2\\overrightarrow{i}\\ + 3\\overrightarrow{j} + \\overrightarrow{k}) + 3 ( 3\\overrightarrow{i}\\ &#8211; \\overrightarrow{j} + \\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=4\\overrightarrow{i}\\ + 6\\overrightarrow{j} + 2\\overrightarrow{k} + 9\\overrightarrow{i}\\ &#8211; 3\\overrightarrow{j} + 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\overrightarrow{a}\\ +\\   3\\overrightarrow{b}= 13\\overrightarrow{i}\\ + 3\\overrightarrow{j} + 5\\overrightarrow{k}\\ \\hspace{5cm}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/pn6b8xr2k-s\" title=\"Vector Introduction (Exercise) - Part - 2\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {3\\ .}\\ 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 {blue} {Soln:}\\ Given\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}= 2\\overrightarrow{i}\\ -\\overrightarrow{j} +  3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}= 5\\overrightarrow{i}\\ + \\overrightarrow{j} &#8211; 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}\\ + \\overrightarrow{j} &#8211; 2\\overrightarrow{k} &#8211; (2\\overrightarrow{i}\\ -\\overrightarrow{j} +  3\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=5\\overrightarrow{i}\\ + \\overrightarrow{j} &#8211; 2\\overrightarrow{k} &#8211; 2\\overrightarrow{i}\\ +\\overrightarrow{j} &#8211;  3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}= 3\\overrightarrow{i} + 2\\overrightarrow{j} &#8211; 5\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB =\\overrightarrow{|AB|} = \\sqrt{(3)^2 + (2)^2 +(-5)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(9 + 4 +25 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{38}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/GJHF5qNQMVo\" title=\"Vector Introduction (Exercise) - Part - 3\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"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<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<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}\\ &#8211; \\overrightarrow{j}- \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{|a|} = \\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\">\\[=\\sqrt{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{|a|}=\\sqrt{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Unit\\  vector\\  along\\ \\overrightarrow{a}=\\frac{\\overrightarrow{a}}{\\overrightarrow{|a|}}= \\frac{2\\overrightarrow{i}\\ &#8211; \\overrightarrow{j}- \\overrightarrow{k}}{\\sqrt{6}}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/Si1YluZRafE\" title=\"Vector Introduction (Exercise) - Part - 4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<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 {blue} {Soln:}\\ Given\\ \\hspace{20cm}\\]<\/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}\\ =\\ 3\\overrightarrow{i}\\ +\\  2\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB} = \\overrightarrow{OB}-\\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 3\\overrightarrow{i}\\ +\\  2\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k}- (\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ +\\  \\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 3\\overrightarrow{i}\\ +\\ 2 \\overrightarrow{j}\\ +\\   3\\overrightarrow{k}\\ -\\ \\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -\\  \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}\\ =\\  2\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\ 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{|AB|} = \\sqrt{(2)^2 + (3)^2 +(2)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(4\\ +\\  9\\ +\\ 4})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\overrightarrow{|AB|}\\ = \\sqrt{17}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Direction\\  ratios\\  are\\ 2,   3,   2  \\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/fIhuJKOsgNE\" title=\"Vector Introduction (Exercise) - Part - 5\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {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=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ Given\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ \\overrightarrow{a}\\ =\\ 2\\overrightarrow{i}\\ + 3\\overrightarrow{j}\\ +\\ 4\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[r =\\overrightarrow{|a|} = \\sqrt{(2)^2 + (3)^2 + (4)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(4\\ +\\  9\\ +\\ 16}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[r =\\sqrt{29}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Direction\\  cosines\\  are \\frac{2}{\\sqrt{29}}\\ ,  \\frac{3}{\\sqrt{29}}\\ ,  \\frac{4}{\\sqrt{29}} \\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/VWE9ts-TnRs\" title=\"Vector Introduction (Exercise) - Part - 06\" 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=\"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{blue} {Soln\\ :}\\ Given\\ \\hspace {18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}\\ =\\ \\overrightarrow{i}\\ -\\  2\\overrightarrow{j}\\ -\\  \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}= 2\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\  3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OC}\\ =\\ -\\ 7 \\overrightarrow{j}\\ -\\ 5\\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}\\ +\\  3\\overrightarrow{k}\\ &#8211; (\\overrightarrow{i}\\ -\\  2\\overrightarrow{j}\\ -\\  \\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=2\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\  3\\overrightarrow{k}\\ &#8211; \\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ +\\  \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}\\ =  \\overrightarrow{i}\\ +\\ 5\\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\">\\[=\\ -\\ 7 \\overrightarrow{j}\\ -\\ 5\\overrightarrow{k}\\ &#8211; (2\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\  3\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ -\\ 7 \\overrightarrow{j}\\ -\\ 5\\overrightarrow{k}\\ &#8211; 2\\overrightarrow{i}\\ &#8211; 3\\overrightarrow{j}\\ &#8211; 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{BC}\\ =\\  -\\ 2\\ \\overrightarrow{i}\\  &#8211; 10\\ \\overrightarrow{j}\\ -\\ 8\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ -\\ 2( \\overrightarrow{i}\\ +\\  5\\overrightarrow{j}\\ +\\ 4\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{BC}\\ =\\ -\\ 2\\ \\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=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/L9korbYjiAo\" title=\"Vector Introduction (Exercise) - Part - 07\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {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{blue} {Soln\\ :}\\ Given\\ \\hspace {18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}\\ =\\ 2\\overrightarrow{i}\\ + 3\\overrightarrow{j}\\ +\\ 4\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}\\ =\\ 3\\overrightarrow{i}\\ +\\ 4\\overrightarrow{j}\\ +\\  2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OC}\\ =\\ 4\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB} = \\overrightarrow{OB}-\\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 3\\overrightarrow{i}\\ +\\ 4\\overrightarrow{j}\\ +\\  2\\overrightarrow{k}\\ &#8211; (2\\overrightarrow{i}\\ + 3\\overrightarrow{j}\\ +\\ 4\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 3\\overrightarrow{i}\\ +\\ 4\\overrightarrow{j}\\ +\\  2\\overrightarrow{k}\\ -\\ 2\\overrightarrow{i}\\ -\\ 3\\overrightarrow{j}\\ -\\ 4\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}\\ =\\  \\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -\\ 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB =\\overrightarrow{|AB|} = \\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\">\\[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\">\\[=\\ 4\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k}\\ -\\ (3\\overrightarrow{i}\\ +\\ 4\\overrightarrow{j}\\ +\\  2\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 4\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k}\\ &#8211; 3\\overrightarrow{i}\\ &#8211; 4\\overrightarrow{j}\\ -\\ 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{BC}=  \\overrightarrow{i} -\\ 2\\overrightarrow{j}\\  +\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[BC =\\overrightarrow{|BC|} = \\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\">\\[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\">\\[=\\ 4\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k}\\ -\\ (2\\overrightarrow{i}\\ + 3\\overrightarrow{j}\\ +\\ 4\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 4\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ +\\ 3\\overrightarrow{k}\\ -\\ 2\\overrightarrow{i}\\ -\\ 3\\overrightarrow{j}\\ -\\  4\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AC}\\ =\\  2\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ -\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AC =\\overrightarrow{|AC|} = \\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\">\\[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\">\\[\\therefore\\ The\\ given\\ triangle\\ is\\ an\\ equilateral\\ triangle\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/na_cn2SxPYI\" title=\"Vector Introduction (Exercise) - Part - 08\" 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\">\\[\\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<div class=\"wp-block-mathml-mathmlblock\">\\[Soln:\\ Given\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OA}\\ =\\ 3\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ -\\ 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OB}\\ =\\ 5\\overrightarrow{i}\\ +\\  \\overrightarrow{j}\\ -\\  3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{OC}\\ =\\ 6\\overrightarrow{i}\\  -\\  \\overrightarrow{j}\\  -\\  \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB} = \\overrightarrow{OB}-\\overrightarrow{OA}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 5\\overrightarrow{i}\\ +\\  \\overrightarrow{j}\\ -\\  3\\overrightarrow{k}\\ &#8211; (3\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ -\\ 2\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 5\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -\\ 3\\overrightarrow{k}\\ -\\  3\\overrightarrow{i}\\ +\\  \\overrightarrow{j}\\ +\\  2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AB}\\ =\\  2\\overrightarrow{i}\\ +\\  2\\overrightarrow{j}\\ -\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB =\\overrightarrow{|AB|} = \\sqrt{(2)^2\\ +\\  (2)^2\\ +\\ (-1)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(4\\ +\\ 4\\ +\\ 1)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\  \\sqrt{9}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB =\\ 3 \\]<\/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}\\  -\\  \\overrightarrow{j}\\  -\\  \\overrightarrow{k}\\ &#8211; (5\\overrightarrow{i}\\ +\\  \\overrightarrow{j}\\ -\\  3\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 6\\overrightarrow{i}\\  -\\  \\overrightarrow{j}\\  -\\  \\overrightarrow{k}\\ -\\  5\\overrightarrow{i}\\ &#8211; \\overrightarrow{j}\\ +\\  3\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{BC}\\ =\\  \\overrightarrow{i}\\ -\\ 2\\overrightarrow{j}\\ +\\ 2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[BC =\\overrightarrow{|BC|} = \\sqrt{(1)^2 + (- 2)^2 +(2)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(1\\ +\\ 4\\ +\\ 4)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\  \\sqrt{9}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[BC =\\ 3\\]<\/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}\\  -\\  \\overrightarrow{j}\\  -\\  \\overrightarrow{k}\\ &#8211; (3\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ -\\ 2\\overrightarrow{k})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 6\\overrightarrow{i}\\  -\\  \\overrightarrow{j}\\  -\\  \\overrightarrow{k}\\ -\\ 3\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ +\\  2\\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\overrightarrow{AC}\\ =\\  3\\overrightarrow{i}\\ +\\ \\overrightarrow{k}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AC =\\overrightarrow{|AC|} = \\sqrt{(3)^2\\ +\\  (0)^2\\ +\\ (1)^2 }\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(9\\ +\\  0\\ +\\  1)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AC = \\sqrt{10}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{AB = BC\\ \\neq\\ AC}\\]<\/div>\n\n\n\n<p>The given triangle is an isosceles triangle.<\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/FVxm_cVLJeY\" title=\"Vector Introduction (Exercise) - Part - 09\" 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=\"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","protected":false},"excerpt":{"rendered":"<p>The given triangle is an isosceles triangle.<\/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":[711788410],"tags":[],"class_list":["post-14838","post","type-post","status-publish","format-standard","hentry","category-vector-introduction-excercise-with-solutions"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>N &#8211; 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