{"id":6235,"date":"2021-10-10T21:56:00","date_gmt":"2021-10-10T16:26:00","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6235"},"modified":"2022-01-16T17:04:47","modified_gmt":"2022-01-16T11:34:47","slug":"coplanar-vectors","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/coplanar-vectors\/","title":{"rendered":"Coplanar Vectors – Definition and Example"},"content":{"rendered":"

Here you will learn definition of coplanar vectors with example and test of coplanarity of four points.<\/p>\n

Let’s begin –<\/p>\n

Coplanar Vectors<\/h2>\n

A system of vectors is said to be coplanar, if their supports are parallel to the same plane.<\/p>\n

Note<\/strong> : Two vectors are coplanar.<\/p>\n

Theorem 1 (Test of Coplanarity of Three vectors)<\/strong><\/p>\n

\n

Let \\(\\vec{a}\\) and \\(\\vec{b}\\) be two given non-zero non-collinear vectors. Then, any vector \\(\\vec{r}\\) coplanar with \\(\\vec{a}\\) and \\(\\vec{b}\\) can be uniquely expressed as \\(\\vec{r}\\) = \\(x\\vec{r}\\) + \\(y\\vec{b}\\), for some scalars x and y.<\/p>\n<\/blockquote>\n

Theorem 2<\/strong> <\/p>\n

\n

The necessary and sufficient condition for three vectors \\(\\vec{a}\\), \\(\\vec{b}\\) and \\(\\vec{c}\\) to be coplanar is that there exist scalars l, m, n not all zero simultaneously such that \\(l\\vec{a} + m\\vec{b} + n\\vec{c}\\) = \\(\\vec{0}\\).<\/p>\n<\/blockquote>\n

Theorem 3<\/strong><\/p>\n

\n

If \\(\\vec{a}\\), \\(\\vec{b}\\) and \\(\\vec{c}\\) are three non-zero non-coplanar vector and x, y, z are three scalars, then \\(x\\vec{a} + y\\vec{b} + z\\vec{c}\\) = \\(\\vec{0}\\) \\(\\implies\\) x = y = z = 0.<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Show that the vectors \\(\\vec{a} – 2\\vec{b} + 3\\vec{c}\\), \\(\\vec{a} – 3\\vec{b} + 5\\vec{c}\\) and \\(-2\\vec{a} + 3\\vec{b} – 4\\vec{c}\\) are coplanar, where \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are non-coplanar vector.<\/p>\n

Solution<\/span><\/strong> : From Theorem 1, <\/p>\n

Three vectors are co-planar if one of the given vectors are expressible as a linear combination of the other two. Let<\/p>\n

\\(\\vec{a} – 2\\vec{b} + 3\\vec{c}\\) = x\\(\\vec{a} – 3\\vec{b} + 5\\vec{c}\\) + y\\(-2\\vec{a} + 3\\vec{b} – 4\\vec{c}\\) for some scalars x and y.<\/p>\n

\\(-2\\vec{a} + 3\\vec{b} – 4\\vec{c}\\) = (x – 2y)\\(\\vec{a}\\) + (-3x + 3y)\\(\\vec{b}\\) + (5x – 4y)\\(\\vec{c}\\)<\/p>\n

\\(\\implies\\) 1 = x – 2y, -2 = -3x + 3y and 3 = 5x – 4y<\/p>\n

Solving first two of these equation, we get<\/p>\n

x = 1\/3, y = -1\/3.<\/p>\n

Clearly, these values of x and y satisfy the third equation.<\/p>\n

Hence, the given vectors are co-planar.<\/p>\n

Test of Coplanarity of Four Points<\/h2>\n

Four points with position vectors \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) and \\(\\vec{d}\\) are co-planar iff there exist scalars x, y z, u not allzero such that \\(x\\vec{a} + y\\vec{b} + z\\vec{c} + u\\vec{d}\\) = \\(\\vec{0}\\), where x + y + z + u =0.<\/p>\n\n\n

\n
Previous – Types of Vectors in Maths<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn definition of coplanar vectors with example and test of coplanarity of four points. Let’s begin – Coplanar Vectors A system of vectors is said to be coplanar, if their supports are parallel to the same plane. Note : Two vectors are coplanar. Theorem 1 (Test of Coplanarity of Three vectors) Let …<\/p>\n

Coplanar Vectors – Definition and Example<\/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":"\nCoplanar Vectors - Definition and Example - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn definition of coplanar vectors with example and test of coplanarity of four points.\" \/>\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\/coplanar-vectors\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Coplanar Vectors - Definition and Example - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"In this post you will learn definition of coplanar vectors with example and test of coplanarity of four points.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/coplanar-vectors\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-10T16:26:00+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-01-16T11:34:47+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=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/coplanar-vectors\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/coplanar-vectors\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Coplanar Vectors – Definition and Example\",\"datePublished\":\"2021-10-10T16:26:00+00:00\",\"dateModified\":\"2022-01-16T11:34:47+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/coplanar-vectors\/\"},\"wordCount\":398,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"articleSection\":[\"Vectors\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/coplanar-vectors\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/coplanar-vectors\/\",\"url\":\"https:\/\/mathemerize.com\/coplanar-vectors\/\",\"name\":\"Coplanar Vectors - Definition and Example - Mathemerize\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-10-10T16:26:00+00:00\",\"dateModified\":\"2022-01-16T11:34:47+00:00\",\"description\":\"In this post you will learn definition of coplanar vectors with example and test of coplanarity of four points.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/coplanar-vectors\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/coplanar-vectors\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/coplanar-vectors\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Coplanar Vectors – Definition and Example\"}]},{\"@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":"Coplanar Vectors - Definition and Example - Mathemerize","description":"In this post you will learn definition of coplanar vectors with example and test of coplanarity of four points.","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\/coplanar-vectors\/","og_locale":"en_US","og_type":"article","og_title":"Coplanar Vectors - Definition and Example - Mathemerize","og_description":"In this post you will learn definition of coplanar vectors with example and test of coplanarity of four points.","og_url":"https:\/\/mathemerize.com\/coplanar-vectors\/","og_site_name":"Mathemerize","article_published_time":"2021-10-10T16:26:00+00:00","article_modified_time":"2022-01-16T11:34:47+00:00","author":"mathemerize","twitter_card":"summary_large_image","twitter_misc":{"Written by":"mathemerize","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathemerize.com\/coplanar-vectors\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/coplanar-vectors\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"Coplanar Vectors – Definition and Example","datePublished":"2021-10-10T16:26:00+00:00","dateModified":"2022-01-16T11:34:47+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/coplanar-vectors\/"},"wordCount":398,"commentCount":0,"publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"articleSection":["Vectors"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathemerize.com\/coplanar-vectors\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/coplanar-vectors\/","url":"https:\/\/mathemerize.com\/coplanar-vectors\/","name":"Coplanar Vectors - Definition and Example - Mathemerize","isPartOf":{"@id":"https:\/\/mathemerize.com\/#website"},"datePublished":"2021-10-10T16:26:00+00:00","dateModified":"2022-01-16T11:34:47+00:00","description":"In this post you will learn definition of coplanar vectors with example and test of coplanarity of four points.","breadcrumb":{"@id":"https:\/\/mathemerize.com\/coplanar-vectors\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathemerize.com\/coplanar-vectors\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathemerize.com\/coplanar-vectors\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathemerize.com\/"},{"@type":"ListItem","position":2,"name":"Coplanar Vectors – Definition and Example"}]},{"@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\/6235"}],"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=6235"}],"version-history":[{"count":9,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/6235\/revisions"}],"predecessor-version":[{"id":9563,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/6235\/revisions\/9563"}],"wp:attachment":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/media?parent=6235"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/categories?post=6235"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/tags?post=6235"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}