{"id":3918,"date":"2021-08-16T20:46:09","date_gmt":"2021-08-16T20:46:09","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3918"},"modified":"2021-10-10T20:00:14","modified_gmt":"2021-10-10T14:30:14","slug":"definition-of-collinear-vectors","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/definition-of-collinear-vectors\/","title":{"rendered":"Definition of Collinear Vectors"},"content":{"rendered":"

Here, you will learn definition of collinear vectors, coplanar vectors, co-initial vectors and test of collinearity of three points.<\/p>\n

Let’s begin –<\/p>\n

Definition of Collinear Vectors<\/h2>\n

Two vectors are said to be collinear if their supports are parallel disregards to their direction. Collinear vectors are also called Parallel vectors<\/strong>. If they have the same direction they are named as like vectors otherwise unlike vectors.<\/p>\n

\n

Symbolically, If \\(\\vec{a}\\) & \\(\\vec{b}\\) are collinear or parallel vectors, then there exists a scalar \\(\\lambda\\) such that \\(\\vec{a}\\) = \\(\\lambda\\vec{b}\\) or, \\(\\vec{b}\\) = \\(\\lambda\\vec{a}\\).<\/p>\n<\/blockquote>\n

Theorem 1 :<\/b><\/p>\n

\n

Two non-zero vectors \\(\\vec{a}\\) & \\(\\vec{b}\\) are collinear iff there exist scalars x, y not both zero such that x\\(\\vec{a}\\) + y\\(\\vec{b}\\) = \\(\\vec{0}\\).<\/p>\n<\/blockquote>\n

Theorem 2 :<\/b><\/p>\n

\n

If \\(\\vec{a}\\) & \\(\\vec{b}\\) are two non-zero non-collinear vectors and x, y are scalars then x\\(\\vec{a}\\) + y\\(\\vec{b}\\) = 0 \\(\\implies\\) x = y = 0.<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : If \\(\\vec{a}\\) and \\(\\vec{b}\\) are non-collinear vectors, find the value of x for which vectors \\(\\vec{\\alpha}\\) = (x – 2)\\(\\vec{a}\\) + \\(\\vec{b}\\) and \\(\\vec{\\beta}\\) = (3 + 2x)\\(\\vec{a}\\) – 2\\(\\vec{b}\\) are collinear.<\/p>\n

Solution<\/span><\/strong> : Since vectors \\(\\vec{\\alpha}\\) and \\(\\vec{\\beta}\\) are collinear. Therefore, there exist scalar \\(\\lambda\\) such that<\/p>\n

\\(\\vec{\\alpha}\\) = \\(\\lambda\\)\\(\\vec{\\beta}\\)<\/p>\n

\\(\\implies\\) (x – 2)\\(\\vec{a}\\) + \\(\\vec{b}\\) = \\(\\lambda\\){(3 + 2x)\\(\\vec{a}\\) – 2\\(\\vec{b}\\)}<\/p>\n

\\(\\implies\\) {x – 2 – \\(\\lambda\\)(3 + 2x)}\\(\\vec{a}\\) + (1 + 2\\(\\lambda\\)\\(\\vec{b}\\) = \\(\\vec{0}\\)<\/p>\n

Now, given \\(\\vec{a}\\) and \\(\\vec{b}\\) are non-collinear.<\/p>\n

Therefore, from theorem 2,<\/p>\n

x – 2 – \\(\\lambda\\)(3 + 2x) = 0 and 1 + 2\\(\\lambda\\) = 0<\/p>\n

x – 2 – \\(\\lambda\\)(3 + 2x) = 0 and \\(\\lambda\\) = \\(-1\\over 2\\)<\/p>\n

x – 2 + \\(1\\over 2\\)(3 + 2x) = 0 \\(\\implies\\) 4x + 1 = 0<\/p>\n

\\(\\implies\\) x = \\(-1\\over 4\\)<\/p>\n

Test of Collinearity of three points in Vectors<\/h2>\n

(a)\u00a0 3 points A B C will be collinear if \\(\\overrightarrow{AB}\\) = \\(\\lambda\\overrightarrow{BC}\\), where \\(\\lambda\\) \\(\\in\\) R.<\/p>\n

(b)\u00a0 Three points A, B, C with position vectors \\(\\vec{a}\\),\\(\\vec{b}\\),\\(\\vec{c}\\) respectively are collinear, if & only if there exist scalars x,y,z not all zero simultaneously such that ; x\\(\\vec{a}\\) + y\\(\\vec{b}\\) + z\\(\\vec{c}\\) = 0, where x + y + z = 0.<\/p>\n

(c)\u00a0 Collinearity can also be checked by first finding the equation of line through two points and satisfy the third point.<\/p>\n\n\n

\n
Next – How to Find the Unit Vector \u2013 Formula for Unit Vector<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – State Polygon Law of Vector Addition<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here, you will learn definition of collinear vectors, coplanar vectors, co-initial vectors and test of collinearity of three points. Let’s begin – Definition of Collinear Vectors Two vectors are said to be collinear if their supports are parallel disregards to their direction. Collinear vectors are also called Parallel vectors. If they have the same direction …<\/p>\n

Definition of Collinear Vectors<\/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":"\nDefinition of Collinear Vectors - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post, you will learn definition of collinear vectors with example and test of collinearity of three points in vectors.\" \/>\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\/definition-of-collinear-vectors\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Definition of Collinear Vectors - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"In this post, you will learn definition of collinear vectors with example and test of collinearity of three points in vectors.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/definition-of-collinear-vectors\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-08-16T20:46:09+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-10-10T14:30:14+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\/definition-of-collinear-vectors\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/definition-of-collinear-vectors\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Definition of Collinear Vectors\",\"datePublished\":\"2021-08-16T20:46:09+00:00\",\"dateModified\":\"2021-10-10T14:30:14+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/definition-of-collinear-vectors\/\"},\"wordCount\":392,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"articleSection\":[\"Vectors\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/definition-of-collinear-vectors\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/definition-of-collinear-vectors\/\",\"url\":\"https:\/\/mathemerize.com\/definition-of-collinear-vectors\/\",\"name\":\"Definition of Collinear Vectors - Mathemerize\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-08-16T20:46:09+00:00\",\"dateModified\":\"2021-10-10T14:30:14+00:00\",\"description\":\"In this post, you will learn definition of collinear vectors with example and test of collinearity of three points in vectors.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/definition-of-collinear-vectors\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/definition-of-collinear-vectors\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/definition-of-collinear-vectors\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Definition of Collinear Vectors\"}]},{\"@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":"Definition of Collinear Vectors - Mathemerize","description":"In this post, you will learn definition of collinear vectors with example and test of collinearity of three points in vectors.","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\/definition-of-collinear-vectors\/","og_locale":"en_US","og_type":"article","og_title":"Definition of Collinear Vectors - Mathemerize","og_description":"In this post, you will learn definition of collinear vectors with example and test of collinearity of three points in vectors.","og_url":"https:\/\/mathemerize.com\/definition-of-collinear-vectors\/","og_site_name":"Mathemerize","article_published_time":"2021-08-16T20:46:09+00:00","article_modified_time":"2021-10-10T14:30:14+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\/definition-of-collinear-vectors\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/definition-of-collinear-vectors\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"Definition of Collinear Vectors","datePublished":"2021-08-16T20:46:09+00:00","dateModified":"2021-10-10T14:30:14+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/definition-of-collinear-vectors\/"},"wordCount":392,"commentCount":0,"publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"articleSection":["Vectors"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathemerize.com\/definition-of-collinear-vectors\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/definition-of-collinear-vectors\/","url":"https:\/\/mathemerize.com\/definition-of-collinear-vectors\/","name":"Definition of Collinear Vectors - Mathemerize","isPartOf":{"@id":"https:\/\/mathemerize.com\/#website"},"datePublished":"2021-08-16T20:46:09+00:00","dateModified":"2021-10-10T14:30:14+00:00","description":"In this post, you will learn definition of collinear vectors with example and test of collinearity of three points in vectors.","breadcrumb":{"@id":"https:\/\/mathemerize.com\/definition-of-collinear-vectors\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathemerize.com\/definition-of-collinear-vectors\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathemerize.com\/definition-of-collinear-vectors\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathemerize.com\/"},{"@type":"ListItem","position":2,"name":"Definition of Collinear Vectors"}]},{"@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\/3918"}],"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=3918"}],"version-history":[{"count":6,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/3918\/revisions"}],"predecessor-version":[{"id":6234,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/3918\/revisions\/6234"}],"wp:attachment":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/media?parent=3918"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/categories?post=3918"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/tags?post=3918"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}