{"id":6039,"date":"2021-10-04T23:31:44","date_gmt":"2021-10-04T18:01:44","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6039"},"modified":"2021-10-04T23:32:22","modified_gmt":"2021-10-04T18:02:22","slug":"scalar-and-vector-examples","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/scalar-and-vector-examples\/","title":{"rendered":"Scalar and Vector Examples"},"content":{"rendered":"

Here you will learn some scalar and vector examples for better understanding of scalar and vector concepts<\/a><\/span>.<\/p>\n\n\n

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\n\n \n

Example 1 : <\/span>Find the vector of magnitude 5 which are perpendicular to the vectors \\(\\vec{a}\\) = \\(2\\hat{i} + \\hat{j} – 3\\hat{k}\\) and \\(\\vec{b}\\) = \\(\\hat{i} – 2\\hat{j} + \\hat{k}\\).<\/p>\n

Solution : <\/span>Unit vectors perpendicular to \\(\\vec{a}\\) & \\(\\vec{b}\\) = \\(\\pm\\)\\(\\vec{a}\\times\\vec{b}\\over |\\vec{a}\\times\\vec{b}|\\)

\n \\(\\therefore\\)   \\(\\vec{a}\\times\\vec{b}\\) = \\(\\begin{vmatrix}\n \t \\hat{i} & \\hat{j} & \\hat{k} \\\\\n 2 & 1 & -3 \\\\\n 1 & 2 & -2 \\\\\n \\end{vmatrix}\\) = \\(-5\\hat{i} – 5\\hat{j} – 5\\hat{k}\\)

\n \\(\\therefore\\)   Unit Vectors = \\(\\pm\\) \\(-5\\hat{i} – 5\\hat{j} – 5\\hat{k}\\over 5\\sqrt{3}\\)

\n Hence the required vectors are \\(\\pm\\) \\(5\\sqrt{3}\\over 3\\)(\\(\\hat{i} + \\hat{j} + \\hat{k}\\))<\/p>\n


\n\n \n

Example 2 : <\/span> If \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are three non zero vectors such that \\(\\vec{a}\\times\\vec{b}\\) = \\(\\vec{c}\\) and \\(\\vec{b}\\times\\vec{c}\\) = \\(\\vec{a}\\), prove that \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are mutually at right angles and |\\(\\vec{b}\\)| = 1 and |\\(\\vec{c}\\)| = |\\(\\vec{a}\\)|<\/p>\n

Solution : <\/span>\\(\\vec{a}\\times\\vec{b}\\) = \\(\\vec{c}\\) and \\(\\vec{b}\\times\\vec{c}\\) = \\(\\vec{a}\\)

\n \\(\\implies\\)   \\(\\vec{c}\\perp\\vec{a}\\) , \\(\\vec{c}\\perp\\vec{b}\\) and \\(\\vec{a}\\perp\\vec{b}\\), \\(\\vec{a}\\perp\\vec{c}\\)

\n \\(\\implies\\)   \\(\\vec{a}\\perp\\vec{b}\\), \\(\\vec{b}\\perp\\vec{c}\\) and \\(\\vec{c}\\perp\\vec{a}\\)

\n \\(\\implies\\)   \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are mutually perpendicular vectors.

\n Again, \\(\\vec{a}\\times\\vec{b}\\) = \\(\\vec{c}\\) and \\(\\vec{b}\\times\\vec{c}\\) = \\(\\vec{a}\\)

\n \\(\\implies\\)   |\\(\\vec{a}\\times\\vec{b}\\)| = |\\(\\vec{c}\\)| and |\\(\\vec{b}\\times\\vec{c}\\)| = |\\(\\vec{a}\\)|

\n \\(\\implies\\)   \\(|\\vec{a}||\\vec{b}|sin{\\pi\\over 2}\\) = |\\(\\vec{c}\\)| and \\(|\\vec{b}||\\vec{c}|sin{\\pi\\over 2}\\) = |\\(\\vec{a}\\)|     (\\(\\because\\) \\(\\vec{a}\\perp\\vec{b}\\) and \\(\\vec{b}\\perp\\vec{c}\\))

\n \\(\\implies\\)   \\(|\\vec{a}||\\vec{b}|\\) = |\\(\\vec{c}\\)| and \\(|\\vec{b}||\\vec{c}|\\) = |\\(\\vec{a}\\)|

\n \\(\\implies\\)   \\({|\\vec{b}|}^2\\) |\\(\\vec{c}\\)| = |\\(\\vec{c}\\)|

\n \\(\\implies\\)   \\({|\\vec{b}|}^2\\) = 1

\n \\(\\implies\\)   \\(|\\vec{b}|\\) = 1

\n putting in \\(|\\vec{a}||\\vec{b}|\\) = |\\(\\vec{c}\\)|

\n \\(\\implies\\)   \\(|\\vec{a}|\\) = |\\(\\vec{c}\\)|\n <\/p>\n


\n\n \n

Example 3 : <\/span>For any three vectors \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) prove that [\\(\\vec{a}\\) + \\(\\vec{b}\\) \\(\\vec{b}\\) + \\(\\vec{c}\\) \\(\\vec{c}\\) + \\(\\vec{a}\\)] = 2[\\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)]<\/p>\n

Solution : <\/span>We have [\\(\\vec{a}\\) + \\(\\vec{b}\\) \\(\\vec{b}\\) + \\(\\vec{c}\\) \\(\\vec{c}\\) + \\(\\vec{a}\\)]

\n = {(\\(\\vec{a}\\) + \\(\\vec{b}\\))\\(\\times\\)(\\(\\vec{b}\\) + \\(\\vec{c}\\))}.(\\(\\vec{c}\\) + \\(\\vec{a}\\))

\n = {\\(\\vec{a}\\)\\(\\times\\)\\(\\vec{b}\\) + \\(\\vec{a}\\)\\(\\times\\)\\(\\vec{c}\\) + \\(\\vec{b}\\)\\(\\times\\)\\(\\vec{b}\\) + \\(\\vec{b}\\)\\(\\times\\)\\(\\vec{c}\\)}.(\\(\\vec{c}\\) + \\(\\vec{a}\\))       {\\(\\vec{b}\\)\\(\\times\\)\\(\\vec{b}\\) = 0}

\n = {\\(\\vec{a}\\)\\(\\times\\)\\(\\vec{b}\\) + \\(\\vec{a}\\)\\(\\times\\)\\(\\vec{c}\\) + \\(\\vec{b}\\)\\(\\times\\)\\(\\vec{c}\\)}.(\\(\\vec{c}\\) + \\(\\vec{a}\\))

\n = (\\(\\vec{a}\\times\\vec{b}\\)).\\(\\vec{c}\\) + (\\(\\vec{a}\\times\\vec{c}\\)).\\(\\vec{c}\\) + (\\(\\vec{b}\\times\\vec{c}\\)).\\(\\vec{c}\\) + (\\(\\vec{a}\\times\\vec{b}\\)).\\(\\vec{a}\\) + (\\(\\vec{a}\\times\\vec{c}\\)).\\(\\vec{a}\\) + (\\(\\vec{b}\\times\\vec{c}\\)).\\(\\vec{a}\\)

\n = [\\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)] + 0 + 0 + 0 + 0 + [\\(\\vec{b}\\) \\(\\vec{c}\\) \\(\\vec{a}\\)]

\n = [\\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)] + [\\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)] = 2[\\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)]\n <\/p>\n
\n

Practice these given scalar and vector examples to test your knowledge on concepts of scalar and vectors.<\/p>\n \n <\/div>\n <\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn some scalar and vector examples for better understanding of scalar and vector concepts. Example 1 : Find the vector of magnitude 5 which are perpendicular to the vectors \\(\\vec{a}\\) = \\(2\\hat{i} + \\hat{j} – 3\\hat{k}\\) and \\(\\vec{b}\\) = \\(\\hat{i} – 2\\hat{j} + \\hat{k}\\). Solution : Unit vectors perpendicular to \\(\\vec{a}\\) & …<\/p>\n

Scalar and Vector Examples<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[33],"tags":[],"yoast_head":"\nScalar and Vector Examples - Mathemerize<\/title>\n<meta name=\"description\" content=\"Solve these Scalar and Vector examples to practice and test your knowledge of Scalar and Vector concepts.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/scalar-and-vector-examples\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Scalar and Vector Examples - 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