{"id":6386,"date":"2021-10-13T18:20:17","date_gmt":"2021-10-13T12:50:17","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6386"},"modified":"2021-10-14T22:49:26","modified_gmt":"2021-10-14T17:19:26","slug":"intersection-of-a-line-and-a-plane","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/intersection-of-a-line-and-a-plane\/","title":{"rendered":"Intersection of a Line and a Plane"},"content":{"rendered":"

Here you will learn how to find intersection of a line and a plane with examples.<\/p>\n

Let’s begin – <\/p>\n

Intersection of a Line and a Plane<\/h2>\n

Let the equation of a line be \\(x – x_1\\over l\\) = \\(y – y_1\\over m\\) = \\(z – z_1\\over n\\) and that of a plane be ax + by + cz + d = 0.<\/p>\n

The coordinates of any point on the line  \\(x – x_1\\over l\\) = \\(y – y_1\\over m\\) = \\(z – z_1\\over n\\) is given <\/p>\n

\\(x – x_1\\over l\\) = \\(y – y_1\\over m\\) = \\(z – z_1\\over n\\) = r (say)<\/p>\n

or, \\((x_1 + lr, y_1 + mr, z + nr)\\)                …………(i)<\/p>\n

If it lies on the plane ax + by + cz + d = 0, then<\/p>\n

\\(a(x_1 + lr)\\) + \\(b(y_1 + mr)\\) + \\(c(z_1 + nr)\\) + d = 0<\/p>\n

\\(\\implies\\) \\((ax_1 + by_1 + cz_1 + d)\\) + r(al + bm + cn) = 0<\/p>\n

\\(\\implies\\) r = -\\((ax_1 + by_1 + cz_1 + d)\\over al + bm + cn\\)<\/p>\n

Substituting the value of r in (i), we obtain the coordinates of the required point of intersection.<\/p>\n

In order to find the coordinates of the point of intersection of a line and a plane, we may use the following algorithm,<\/p>\n

Algorithm :<\/strong><\/p>\n

\n

1). Write the coordinates of any point on the line in terms of some parameters r (say).<\/p>\n

2). Substitute these coordinates in the equation of the plane to obtain the value of r.<\/p>\n

3). Put the value of r in the coordinates of the point in step 1.<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Find the coordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the XY-plane.<\/p>\n

Solution<\/span><\/strong> : The equation of the line passing through A and B is<\/p>\n

\\(x – 3\\over 5 – 3\\) = \\(y – 4\\over 1 – 4\\) = \\(z – 1\\over 6 – 1\\)<\/p>\n

or,  \\(x – 3\\over 2\\) = \\(y – 4\\over -3\\) = \\(z – 1\\over 5\\)<\/p>\n

The coordinates of any point on this line are given by<\/p>\n

\\(x – 3\\over 2\\) = \\(y – 4\\over -3\\) = \\(z – 1\\over 5\\) = \\(\\lambda\\) <\/p>\n

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

So, \\((2\\lambda + 3, -3\\lambda + 4, 5\\lambda + 1)\\) are coordinates of any point on the line passing through A and B. If it lies on XY-plane i.e z = 0.Then.<\/p>\n

\\(5\\lambda + 1\\) = 0 \\(\\implies\\) \\(\\lambda\\) = -\\(1\\over 5\\)<\/p>\n

Thus, the coordinates of the required point are (13\/5, 23\/5, 0).<\/p>\n\n\n

\n
Next – Equation of Plane Containing Two Lines<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Angle Between a Line and a Plane<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn how to find intersection of a line and a plane with examples. Let’s begin –  Intersection of a Line and a Plane Let the equation of a line be \\(x – x_1\\over l\\) = \\(y – y_1\\over m\\) = \\(z – z_1\\over n\\) and that of a plane be ax + …<\/p>\n

Intersection of a Line and a Plane<\/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":"\nIntersection of a Line and a Plane - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to find intersection of a line and a plane with examples.\" \/>\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\/intersection-of-a-line-and-a-plane\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Intersection of a Line and a Plane - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"In this post you will learn how to find intersection of a line and a plane with examples.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/intersection-of-a-line-and-a-plane\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-13T12:50:17+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-10-14T17:19:26+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\/intersection-of-a-line-and-a-plane\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/intersection-of-a-line-and-a-plane\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Intersection of a Line and a Plane\",\"datePublished\":\"2021-10-13T12:50:17+00:00\",\"dateModified\":\"2021-10-14T17:19:26+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/intersection-of-a-line-and-a-plane\/\"},\"wordCount\":397,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"articleSection\":[\"Vectors\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/intersection-of-a-line-and-a-plane\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/intersection-of-a-line-and-a-plane\/\",\"url\":\"https:\/\/mathemerize.com\/intersection-of-a-line-and-a-plane\/\",\"name\":\"Intersection of a Line and a Plane - Mathemerize\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-10-13T12:50:17+00:00\",\"dateModified\":\"2021-10-14T17:19:26+00:00\",\"description\":\"In this post you will learn how to find intersection of a line and a plane with examples.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/intersection-of-a-line-and-a-plane\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/intersection-of-a-line-and-a-plane\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/intersection-of-a-line-and-a-plane\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Intersection of a Line and a Plane\"}]},{\"@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":"Intersection of a Line and a Plane - Mathemerize","description":"In this post you will learn how to find intersection of a line and a plane with examples.","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\/intersection-of-a-line-and-a-plane\/","og_locale":"en_US","og_type":"article","og_title":"Intersection of a Line and a Plane - Mathemerize","og_description":"In this post you will learn how to find intersection of a line and a plane with examples.","og_url":"https:\/\/mathemerize.com\/intersection-of-a-line-and-a-plane\/","og_site_name":"Mathemerize","article_published_time":"2021-10-13T12:50:17+00:00","article_modified_time":"2021-10-14T17:19:26+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\/intersection-of-a-line-and-a-plane\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/intersection-of-a-line-and-a-plane\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"Intersection of a Line and a Plane","datePublished":"2021-10-13T12:50:17+00:00","dateModified":"2021-10-14T17:19:26+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/intersection-of-a-line-and-a-plane\/"},"wordCount":397,"commentCount":0,"publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"articleSection":["Vectors"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathemerize.com\/intersection-of-a-line-and-a-plane\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/intersection-of-a-line-and-a-plane\/","url":"https:\/\/mathemerize.com\/intersection-of-a-line-and-a-plane\/","name":"Intersection of a Line and a Plane - Mathemerize","isPartOf":{"@id":"https:\/\/mathemerize.com\/#website"},"datePublished":"2021-10-13T12:50:17+00:00","dateModified":"2021-10-14T17:19:26+00:00","description":"In this post you will learn how to find intersection of a line and a plane with examples.","breadcrumb":{"@id":"https:\/\/mathemerize.com\/intersection-of-a-line-and-a-plane\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathemerize.com\/intersection-of-a-line-and-a-plane\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathemerize.com\/intersection-of-a-line-and-a-plane\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathemerize.com\/"},{"@type":"ListItem","position":2,"name":"Intersection of a Line and a Plane"}]},{"@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\/6386"}],"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=6386"}],"version-history":[{"count":5,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/6386\/revisions"}],"predecessor-version":[{"id":6463,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/6386\/revisions\/6463"}],"wp:attachment":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/media?parent=6386"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/categories?post=6386"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/tags?post=6386"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}