{"id":2612,"date":"2021-07-13T19:45:44","date_gmt":"2021-07-13T19:45:44","guid":{"rendered":"https:\/\/mathemerize.com\/?p=2612"},"modified":"2021-10-08T01:28:13","modified_gmt":"2021-10-07T19:58:13","slug":"cross-product-of-vectors-formula","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/","title":{"rendered":"Cross Product of Vectors Formula [ Vector Product ]"},"content":{"rendered":"

Cross Product of Vectors Formula :<\/h2>\n

Let \\(\\vec{a}\\) & \\(\\vec{b}\\) are two vectors & \\(\\theta\\) is the angle between them, then cross product of vectors formula is,<\/p>\n

\n

\\(\\vec{a}\\) \\(\\times\\) \\(\\vec{b}\\) = |\\(\\vec{a}\\)||\\(\\vec{b}\\)|sin\\(\\theta\\)\\(\\hat{n}\\)<\/p>\n

where \\(\\hat{n}\\) is the unit vector perpendicular to both \\(\\vec{a}\\) & \\(\\vec{b}\\).<\/p>\n<\/blockquote>\n

Properties of Vector Cross Product :<\/h2>\n

(i) \\(\\vec{a}\\) \\(\\times\\) \\(\\vec{b}\\) = \\(\\vec{0}\\) \\(\\iff\\) \\(\\vec{a}\\) & \\(\\vec{b}\\) are parallel(Collinear) (\\(\\vec{a}\\) \\(\\ne\\) 0, \\(\\vec{b}\\) \\(\\ne\\) 0) i.e. \\(\\vec{a}\\) = K\\(\\vec{b}\\), where K is a scalar.<\/p>\n

(ii) \\(\\vec{a}\\) \\(\\times\\) \\(\\vec{b}\\) \\(\\ne\\) \\(\\vec{b}\\) \\(\\times\\) \\(\\vec{a}\\)  (not commutative)<\/p>\n

(iii) m(\\(\\vec{a}\\)) \\(\\times\\) \\(\\vec{b}\\) = \\(\\vec{a}\\) \\(\\times\\) (m\\(\\vec{b}\\)) = m(\\(\\vec{a}\\) \\(\\times\\) \\(\\vec{b}\\)) where m is a scalar.<\/p>\n

(iv) \\(\\vec{a}\\) \\(\\times\\) (\\(\\vec{b}\\) + \\(\\vec{c}\\))  (distributive over addition)<\/p>\n

(v)  \\(\\hat{i}\\) \\(\\times\\) \\(\\hat{i}\\) = \\(\\hat{j}\\) \\(\\times\\) \\(\\hat{j}\\) = \\(\\hat{k}\\) \\(\\times\\) \\(\\hat{k}\\) = 0<\/p>\n

(vi) \\(\\hat{i}\\) \\(\\times\\) \\(\\hat{j}\\) = \\(\\hat{k}\\), \\(\\hat{j}\\) \\(\\times\\) \\(\\hat{k}\\) = \\(\\hat{i}\\), \\(\\hat{k}\\) \\(\\times\\) \\(\\hat{i}\\) = \\(\\hat{j}\\)<\/p>\n

(vii) If \\(\\vec{a}\\) = \\(a_1\\hat{i}\\) + \\(a_2\\hat{j}\\) + \\(a_3\\hat{k}\\)  \\(\\vec{b}\\) = \\(b_1\\hat{i}\\) + \\(b_2\\hat{j}\\) + \\(b_3\\hat{k}\\), then \\(\\vec{a}\\) \\(\\times\\) \\(\\vec{b}\\) = \\(\\begin{vmatrix}
\\hat{i} & \\hat{j} & \\hat{k} \\\\
a_1 & a_2 & a_3 \\\\
b_1 & b_2 & b_3 \\\\
\\end{vmatrix}\\)<\/p>\n\n\n

Example : <\/span>Find \\(\\vec{a}\\) \\(\\times\\) \\(\\vec{b}\\), if \\(\\vec{a}\\) = \\(2\\hat{i} +\\hat{k}\\) and \\(\\vec{b}\\) = \\(\\hat{i} +\\hat{j} + \\hat{k}\\)<\/p>\n

Solution : <\/span>We have, \\(\\vec{a}\\) = \\(2\\hat{i} +\\hat{k}\\) and \\(\\vec{b}\\) = \\(\\hat{i} +\\hat{j} + \\hat{k}\\)

\n \\(\\therefore\\)   \\(\\vec{a}\\times\\vec{b}\\) = \\(\\begin{vmatrix}\n \\hat{i} & \\hat{j} & \\hat{k} \\\\\n 2 & 0 & 1 \\\\\n 1 & 1 & 1 \\\\\n \\end{vmatrix}\\) = \\(-1\\hat{i} – 1\\hat{j} + 2\\hat{k}\\)

\n <\/p>\n\n\n

Vectors Normal to the Plane of Two Given Vectors :<\/h2>\n

Let \\(\\vec{a}\\) & \\(\\vec{b}\\) be two non-zero, non-parallel vectors and let \\(\\theta\\) be the angle between them.<\/p>\n

\\(\\vec{a}\\) \\(\\times\\) \\(\\vec{b}\\) = |\\(\\vec{a}\\)||\\(\\vec{b}\\)|sin\\(\\theta\\)\\(\\hat{n}\\),<\/p>\n

where \\(\\hat{n}\\) is the unit vector perpendicular to plane of \\(\\vec{a}\\) & \\(\\vec{b}\\) such that \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\hat{n}\\) form a right-handed system.<\/p>\n

\\(\\therefore\\) \\(\\vec{a}\\) \\(\\times\\) \\(\\vec{b}\\) = | \\(\\vec{a}\\) \\(\\times\\) \\(\\vec{b}\\)|\\(\\hat{n}\\)<\/p>\n

\\(\\implies\\) \\(\\hat{n}\\) = \\(\\vec{a}\\times\\vec{b}\\over |\\vec{a}\\times\\vec{b}|\\)<\/p>\n

\n

Thus, \\(\\vec{a}\\times\\vec{b}\\over |\\vec{a}\\times\\vec{b}|\\) is a unit vector perpendicular to the plane of \\(\\vec{a}\\) and \\(\\vec{b}\\).<\/p>\n

Note that -\\(\\vec{a}\\times\\vec{b}\\over |\\vec{a}\\times\\vec{b}|\\) is also a unit vector perpendicular to the plane of \\(\\vec{a}\\) and \\(\\vec{b}\\).<\/p>\n

Vector of magnitude ‘\\(\\lambda\\)’ normal to the plane of \\(\\vec{a}\\) and \\(\\vec{b}\\) are given by \\(\\pm\\)\\(\\lambda(\\vec{a}\\times\\vec{b})\\over |\\vec{a}\\times\\vec{b}|\\).<\/p>\n<\/blockquote>\n\n\n

Example : <\/span>Find a unit vector perpendicular to both the vectors \\(\\hat{i} -2\\hat{j} + 3\\hat{k}\\) and \\(\\hat{i} + 2\\hat{j} – \\hat{k}\\).<\/p>\n

Solution : <\/span>Let \\(\\vec{a}\\) = \\(\\hat{i} -2\\hat{j} + 3\\hat{k}\\) and \\(\\vec{b}\\) = \\(\\hat{i} + 2\\hat{j} – \\hat{k}\\)

\n\\(\\therefore\\)   \\(\\vec{a}\\times\\vec{b}\\) = \\(\\begin{vmatrix}\n \\hat{i} & \\hat{j} & \\hat{k} \\\\\n 1 & -2 & 3 \\\\\n 1 & 2 & -1 \\\\\n \\end{vmatrix}\\) = \\((2-6)\\hat{i} – (-1-3)\\hat{j} + (2+2)\\hat{k}\\)

\n= \\(-4\\hat{i} + 4\\hat{j} + 4\\hat{k}\\)

\n\\(\\therefore\\) | \\(\\vec{a}\\) \\(\\times\\) \\(\\vec{b}\\)| = \\(\\sqrt{(-4)^2+4^2+4^2}\\) = 4\\(\\sqrt{3}\\)

\nHence, a unit vector perpendicular to vectors \\(\\vec{a}\\) and \\(\\vec{b}\\) is given by

\n\\(\\hat{n}\\) = \\(\\vec{a}\\times\\vec{b}\\over |\\vec{a}\\times\\vec{b}|\\) = \\(-4\\hat{i} + 4\\hat{j} + 4\\hat{k}\\over 4\\sqrt{3}\\) = \\(1\\over \\sqrt{3}\\)(\\(-\\hat{i}+\\hat{j}+\\hat{k}\\)).\n <\/p>\n\n\n

Hope you learnt cross product of vectors formula, learn more concepts of vectors and practice more questions to get ahead in the competition. Good luck!<\/p>\n\n\n

\n
Next – What is Scalar Triple Product \u2013 Properties and Examples<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – What is Dot Product of Two Vectors ? <\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Cross Product of Vectors Formula : Let \\(\\vec{a}\\) & \\(\\vec{b}\\) are two vectors & \\(\\theta\\) is the angle between them, then cross product of vectors formula is, \\(\\vec{a}\\) \\(\\times\\) \\(\\vec{b}\\) = |\\(\\vec{a}\\)||\\(\\vec{b}\\)|sin\\(\\theta\\)\\(\\hat{n}\\) where \\(\\hat{n}\\) is the unit vector perpendicular to both \\(\\vec{a}\\) & \\(\\vec{b}\\). Properties of Vector Cross Product : (i) \\(\\vec{a}\\) \\(\\times\\) \\(\\vec{b}\\) = …<\/p>\n

Cross Product of Vectors Formula [ Vector Product ]<\/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":"\nCross Product of Vectors Formula [ Vector Product ] - Mathemerize<\/title>\n<meta name=\"description\" content=\"Learn what is cross product and cross product of vectors formula and vectors normal to the plane of two given vectors with examples here.\" \/>\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\/cross-product-of-vectors-formula\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Cross Product of Vectors Formula [ Vector Product ] - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"Learn what is cross product and cross product of vectors formula and vectors normal to the plane of two given vectors with examples here.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-07-13T19:45:44+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-10-07T19:58:13+00:00\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Cross Product of Vectors Formula [ Vector Product ]\",\"datePublished\":\"2021-07-13T19:45:44+00:00\",\"dateModified\":\"2021-10-07T19:58:13+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/\"},\"wordCount\":687,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"articleSection\":[\"Vectors\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/\",\"url\":\"https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/\",\"name\":\"Cross Product of Vectors Formula [ Vector Product ] - Mathemerize\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-07-13T19:45:44+00:00\",\"dateModified\":\"2021-10-07T19:58:13+00:00\",\"description\":\"Learn what is cross product and cross product of vectors formula and vectors normal to the plane of two given vectors with examples here.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Cross Product of Vectors Formula [ Vector Product ]\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - Study Math Online\",\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathemerize.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathemerize.com\/#organization\",\"name\":\"Mathemerize\",\"url\":\"https:\/\/mathemerize.com\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"contentUrl\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"width\":140,\"height\":96,\"caption\":\"Mathemerize\"},\"image\":{\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.instagram.com\/mathemerize\/\"]},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\",\"name\":\"mathemerize\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"caption\":\"mathemerize\"},\"sameAs\":[\"https:\/\/mathemerize.com\"],\"url\":\"https:\/\/mathemerize.com\/author\/mathemerize\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Cross Product of Vectors Formula [ Vector Product ] - Mathemerize","description":"Learn what is cross product and cross product of vectors formula and vectors normal to the plane of two given vectors with examples here.","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:\/\/mathemerize.com\/cross-product-of-vectors-formula\/","og_locale":"en_US","og_type":"article","og_title":"Cross Product of Vectors Formula [ Vector Product ] - Mathemerize","og_description":"Learn what is cross product and cross product of vectors formula and vectors normal to the plane of two given vectors with examples here.","og_url":"https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/","og_site_name":"Mathemerize","article_published_time":"2021-07-13T19:45:44+00:00","article_modified_time":"2021-10-07T19:58:13+00:00","author":"mathemerize","twitter_card":"summary_large_image","twitter_misc":{"Written by":"mathemerize","Est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"Cross Product of Vectors Formula [ Vector Product ]","datePublished":"2021-07-13T19:45:44+00:00","dateModified":"2021-10-07T19:58:13+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/"},"wordCount":687,"commentCount":0,"publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"articleSection":["Vectors"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/","url":"https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/","name":"Cross Product of Vectors Formula [ Vector Product ] - Mathemerize","isPartOf":{"@id":"https:\/\/mathemerize.com\/#website"},"datePublished":"2021-07-13T19:45:44+00:00","dateModified":"2021-10-07T19:58:13+00:00","description":"Learn what is cross product and cross product of vectors formula and vectors normal to the plane of two given vectors with examples here.","breadcrumb":{"@id":"https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathemerize.com\/cross-product-of-vectors-formula\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathemerize.com\/"},{"@type":"ListItem","position":2,"name":"Cross Product of Vectors Formula [ Vector Product ]"}]},{"@type":"WebSite","@id":"https:\/\/mathemerize.com\/#website","url":"https:\/\/mathemerize.com\/","name":"Mathemerize","description":"Maths Tutorials - Study Math Online","publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathemerize.com\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/mathemerize.com\/#organization","name":"Mathemerize","url":"https:\/\/mathemerize.com\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/","url":"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1","contentUrl":"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1","width":140,"height":96,"caption":"Mathemerize"},"image":{"@id":"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.instagram.com\/mathemerize\/"]},{"@type":"Person","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df","name":"mathemerize","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g","caption":"mathemerize"},"sameAs":["https:\/\/mathemerize.com"],"url":"https:\/\/mathemerize.com\/author\/mathemerize\/"}]}},"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/2612"}],"collection":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/comments?post=2612"}],"version-history":[{"count":27,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/2612\/revisions"}],"predecessor-version":[{"id":6158,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/2612\/revisions\/6158"}],"wp:attachment":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/media?parent=2612"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/categories?post=2612"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/tags?post=2612"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}