{"id":6301,"date":"2021-10-12T16:50:36","date_gmt":"2021-10-12T11:20:36","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6301"},"modified":"2021-10-12T22:49:59","modified_gmt":"2021-10-12T17:19:59","slug":"equation-of-a-line-perpendicular-to-a-line","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/equation-of-a-line-perpendicular-to-a-line\/","title":{"rendered":"Equation of a Line Perpendicular to a Line"},"content":{"rendered":"

Here you will learn what is the equation of a line perpendicular to a line with proof and examples.<\/p>\n

Let’s begin –<\/p>\n

Equation of a Line Perpendicular to a Line<\/h2>\n
\n

The equation of a line perpendicular to a given line ax + by + c = 0 is<\/p>\n

bx – ay + \\(\\lambda\\) = 0,<\/p>\n

where \\(\\lambda\\) is constant.<\/p>\n<\/blockquote>\n

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

\n

Let \\(m_1\\) be the slope of the given line and \\(m_2\\) be the slope of a line perpendicular to the given line. Then,<\/p>\n

\\(m_1\\) = -\\(a\\over b\\)<\/p>\n

But, \\(m_1m_2\\) = -1 for perpendicular lines<\/p>\n

\\(\\implies\\) \\(m_2\\) = -\\(1\\over m_1\\) = \\(b\\over a\\)<\/p>\n

Let \\(c_2\\) be the y-intercept of the required line. Then, its equation is<\/p>\n

y = \\(m_2\\)x + \\(c_2\\)<\/p>\n

y = \\(b\\over a\\)x + \\(c_2\\)<\/p>\n

\\(\\implies\\) bx – ay + a\\(c_2\\) = 0<\/p>\n

\\(\\implies\\) bx – ay + \\(\\lambda\\), where \\(\\lambda\\) = a\\(c_2\\) = constant.<\/p>\n<\/blockquote>\n

To write a line perpendicular to a given line we proceed as follows<\/strong> :<\/p>\n

\n

1). Interchanging  x and y.<\/p>\n

2). If the coefficients of  x and y in the given equation are of the same sign make them of opposite sign and if the coefficients are of opposite signs make them of the same sign.<\/p>\n

3). Replace the given constant by a new constant \\(\\lambda\\) which is determined by the given condition.<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Find the equation of the straight line that passes through the point (3, 4) and perpendicular to the line 3x + 2y + 5 = 0.<\/p>\n

Solution<\/strong><\/span> : The line perpendicular to 3x + 2y + 5 = 0 is<\/p>\n

2x – 3y + \\(\\lambda\\) = 0                          ……………(i)<\/p>\n

This passes through the point (3, 4)<\/p>\n

\\(\\therefore\\) 3 \\(\\times\\) 2 – 3 \\(\\times\\) 4 + \\(\\lambda\\) = 0 \\(\\implies\\) \\(\\lambda\\) = 6 <\/p>\n

Putting \\(\\lambda\\) = 6 in equation (i), we get<\/p>\n

2x – 3y + 6 = 0, which is the required equation.<\/p>\n\n\n

\n
Next – Equation of Line Parallel to a Line<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Normal Form of a Line Equation<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is the equation of a line perpendicular to a line with proof and examples. Let’s begin – Equation of a Line Perpendicular to a Line The equation of a line perpendicular to a given line ax + by + c = 0 is bx – ay + \\(\\lambda\\) = 0, …<\/p>\n

Equation of a Line Perpendicular to a Line<\/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":[32],"tags":[],"yoast_head":"\nEquation of a Line Perpendicular to a Line - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is the equation of a line perpendicular to a line with proof and 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\/equation-of-a-line-perpendicular-to-a-line\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Equation of a Line Perpendicular to a Line - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"In this post you will learn what is the equation of a line perpendicular to a line with proof and examples.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/equation-of-a-line-perpendicular-to-a-line\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-12T11:20:36+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-10-12T17:19:59+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\/equation-of-a-line-perpendicular-to-a-line\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/equation-of-a-line-perpendicular-to-a-line\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Equation of a Line Perpendicular to a Line\",\"datePublished\":\"2021-10-12T11:20:36+00:00\",\"dateModified\":\"2021-10-12T17:19:59+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/equation-of-a-line-perpendicular-to-a-line\/\"},\"wordCount\":304,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"articleSection\":[\"Straight Line\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/equation-of-a-line-perpendicular-to-a-line\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/equation-of-a-line-perpendicular-to-a-line\/\",\"url\":\"https:\/\/mathemerize.com\/equation-of-a-line-perpendicular-to-a-line\/\",\"name\":\"Equation of a Line Perpendicular to a Line - Mathemerize\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-10-12T11:20:36+00:00\",\"dateModified\":\"2021-10-12T17:19:59+00:00\",\"description\":\"In this post you will learn what is the equation of a line perpendicular to a line with proof and examples.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/equation-of-a-line-perpendicular-to-a-line\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/equation-of-a-line-perpendicular-to-a-line\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/equation-of-a-line-perpendicular-to-a-line\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Equation of a Line Perpendicular to a Line\"}]},{\"@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":"Equation of a Line Perpendicular to a Line - Mathemerize","description":"In this post you will learn what is the equation of a line perpendicular to a line with proof and 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\/equation-of-a-line-perpendicular-to-a-line\/","og_locale":"en_US","og_type":"article","og_title":"Equation of a Line Perpendicular to a Line - Mathemerize","og_description":"In this post you will learn what is the equation of a line perpendicular to a line with proof and examples.","og_url":"https:\/\/mathemerize.com\/equation-of-a-line-perpendicular-to-a-line\/","og_site_name":"Mathemerize","article_published_time":"2021-10-12T11:20:36+00:00","article_modified_time":"2021-10-12T17:19:59+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\/equation-of-a-line-perpendicular-to-a-line\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/equation-of-a-line-perpendicular-to-a-line\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"Equation of a Line Perpendicular to a Line","datePublished":"2021-10-12T11:20:36+00:00","dateModified":"2021-10-12T17:19:59+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/equation-of-a-line-perpendicular-to-a-line\/"},"wordCount":304,"commentCount":0,"publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"articleSection":["Straight Line"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathemerize.com\/equation-of-a-line-perpendicular-to-a-line\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/equation-of-a-line-perpendicular-to-a-line\/","url":"https:\/\/mathemerize.com\/equation-of-a-line-perpendicular-to-a-line\/","name":"Equation of a Line Perpendicular to a Line - Mathemerize","isPartOf":{"@id":"https:\/\/mathemerize.com\/#website"},"datePublished":"2021-10-12T11:20:36+00:00","dateModified":"2021-10-12T17:19:59+00:00","description":"In this post you will learn what is the equation of a line perpendicular to a line with proof and examples.","breadcrumb":{"@id":"https:\/\/mathemerize.com\/equation-of-a-line-perpendicular-to-a-line\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathemerize.com\/equation-of-a-line-perpendicular-to-a-line\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathemerize.com\/equation-of-a-line-perpendicular-to-a-line\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathemerize.com\/"},{"@type":"ListItem","position":2,"name":"Equation of a Line Perpendicular to a Line"}]},{"@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\/6301"}],"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=6301"}],"version-history":[{"count":9,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/6301\/revisions"}],"predecessor-version":[{"id":6359,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/6301\/revisions\/6359"}],"wp:attachment":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/media?parent=6301"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/categories?post=6301"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/tags?post=6301"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}