{"id":6110,"date":"2021-10-07T17:43:47","date_gmt":"2021-10-07T12:13:47","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6110"},"modified":"2021-10-09T00:40:35","modified_gmt":"2021-10-08T19:10:35","slug":"cartesian-equation-of-a-line","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/cartesian-equation-of-a-line\/","title":{"rendered":"Cartesian Equation of a Line"},"content":{"rendered":"

Here you will learn cartesian equation of line in 3d passing through a fixed point and passing through two points.<\/p>\n

Let’s begin –<\/p>\n

Cartesian Equation of a Line<\/h2>\n

The cartesian equations of a straight line passing through a fixed point \\((x_1, y_1, z_1)\\) having direction ratios proportional to a, b, c is given by<\/p>\n

\n

\\(x – x_1\\over a\\) = \\(y – y_1\\over b\\) = \\(z – z_1\\over c\\)<\/p>\n<\/blockquote>\n

Remark 1<\/strong> : The above form of a line is known as the symmetrical form of a line.<\/p>\n

Remark 2<\/strong> : The parametric equations of the line \\(x – x_1\\over a\\) = \\(y – y_1\\over b\\) = \\(z – z_1\\over c\\) are<\/p>\n

\n

x = \\(x_1 + a\\lambda\\), y = \\(y_1 + b\\lambda\\), z = \\(z_1 + c\\lambda\\), where \\(\\lambda\\) is the parameter.<\/p>\n<\/blockquote>\n

Remark 3<\/strong> : The coordinates of any point on the line \\(x – x_1\\over a\\) = \\(y – y_1\\over b\\) = \\(z – z_1\\over c\\) are<\/p>\n

\n

(\\(x_1 + a\\lambda\\),  \\(y_1 + b\\lambda\\), \\(z_1 + c\\lambda\\)), where \\(\\lambda\\) \\(\\in\\) R.<\/p>\n<\/blockquote>\n

Remark 4<\/strong> : Since the direction cosines of a line are also its direcion ratios. Therefore, equations of a line passing through \\((x_1, y_1, z_1)\\) and having direction cosines l, m, n are<\/p>\n

\n

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

Remark 5<\/strong> : Since x, y and z-axes passes through the origin and have direction cosines 1, 0, 0; 0, 1, 0 and 0, 0, 1 respectively. Therefore, their equations are<\/p>\n

\n

x-axis : \\(x – 0\\over 1\\) = \\(y – 0\\over 0\\) = \\(z – 0\\over 0\\) or, y = 0 and z = 0<\/p>\n

y-axis : \\(x – 0\\over 0\\) = \\(y – 0\\over 1\\) = \\(z – 0\\over 0\\) or, x = 0 and z = 0<\/p>\n

z-axis : \\(x – 0\\over 0\\) = \\(y – 0\\over 0\\) = \\(z – 0\\over 1\\) or, y = 0 and y = 0<\/p>\n<\/blockquote>\n

Cartesian Equation of a Line Passing Through Two Points<\/strong><\/h4>\n

The Cartesian equation of aline passing through two given points \\((x_1, y_1, z_1)\\) and \\((x_2, y_2, z_2)\\) is given by<\/p>\n

\n

\\(x – x_1\\over x_2 – x_1\\) = \\(y – y_1\\over y_2 – y_1\\) = \\(z – z_1\\over z – z_1\\)<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : Find the cartesian equation of line passing through A(3, 4, -7) and B(1, -1, 6).<\/p>\n

Solution<\/strong><\/span> : We have, A(3, 4, -7) and B(1, -1, 6)<\/p>\n

The cartesian equation of line passing through two points is<\/p>\n

\\(x – x_1\\over x_2 – x_1\\) = \\(y – y_1\\over y_2 – y_1\\) = \\(z – z_1\\over z – z_1\\)<\/p>\n

= \\(x – 3\\over -2\\) = \\(y – 4\\over -5\\) = \\(z + 7\\over 13\\)<\/p>\n\n\n

\n
Next – Reduction of Cartesian Form of Line to Vector Form and Vice-Versa<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Equation of a Line in Vector Form<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn cartesian equation of line in 3d passing through a fixed point and passing through two points. Let’s begin – Cartesian Equation of a Line The cartesian equations of a straight line passing through a fixed point \\((x_1, y_1, z_1)\\) having direction ratios proportional to a, b, c is given by \\(x …<\/p>\n

Cartesian Equation of 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":[33],"tags":[],"yoast_head":"\nCartesian Equation of a Line - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn cartesian equation of line in 3d passing through a fixed point and passing through two 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\/cartesian-equation-of-a-line\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Cartesian Equation of a Line - 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