{"id":6366,"date":"2021-10-13T17:54:09","date_gmt":"2021-10-13T12:24:09","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6366"},"modified":"2021-10-14T22:43:51","modified_gmt":"2021-10-14T17:13:51","slug":"equation-of-plane-passing-through-three-points","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/equation-of-plane-passing-through-three-points\/","title":{"rendered":"Equation of Plane Passing Through Three Points"},"content":{"rendered":"

Here you will learn the equation of plane passing through three points with example.<\/p>\n

Let’s begin –<\/p>\n

Equation of a Plane Passing Through a Given Point<\/h2>\n
\n

The general equation of a plane passing through a point \\((x_1, y_1, z_1\\)) is <\/p>\n

\\(a(x – x_1) + b(y – y_1) + c(z – z_1)\\) = 0, where a, b and c are constants.<\/p>\n<\/blockquote>\n

Now, In order to find the equation of plane passing through three given points \\((x_1, y_1, z_1\\)), \\((x_2, y_2, z_2\\)) and \\((x_3, y_3, z_3\\)), we may use the following algorithm.<\/p>\n

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

\n

1)<\/strong>. Write the equation of a plane passing through \\((x_1, y_1, z_1\\)) as<\/p>\n

\\(a(x – x_1) + b(y – y_1) + c(z – z_1)\\) = 0                     ………..(i)<\/p>\n

2)<\/strong>. If the plane (i) passes through \\((x_2, y_2, z_2\\)) and \\((x_3, y_3, z_3\\)), then<\/p>\n

\\(a(x_2 – x_1) + b(y_2 – y_1) + c(z_2 – z_1)\\) = 0                     ………..(ii)<\/p>\n

\\(a(x_3 – x_1) + b(y_3 – y_1) + c(z_3 – z_1)\\) = 0                     ………..(iii)<\/p>\n

3)<\/strong>. Solve equation (ii) and (iii), obtained in step 2, by cross-multiplication.<\/p>\n

4)<\/strong>. Substituting the values of a, b, c, obtained in step 3, in equation (i) in step 1 to get the required plane.<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Find the equation of the plane through the points A(2, 2, -1), B(3, 4, 2) and C(7, 0 , 6).<\/p>\n

Solution<\/span><\/strong> : The general equation of a plane passing through (2, 2, -1) is<\/p>\n

a(x – 2) + b(y – 2) + c(z + 1) = 0                                 ……………..(i)<\/p>\n

It will pass through B(3, 4, 2) and C(7, 0 , 6), if<\/p>\n

a(3 – 2) + b(4 – 2) + c(2 + 1) = 0 \\(\\implies\\) a + 2b + 3c = 0                           ………..(ii)<\/p>\n

and, a(7 – 2) + b(0 – 2) + c(6 + 1) = 0 \\(\\implies\\) 5a – 2b + 7c = 0                 …………(iii)<\/p>\n

Solving (ii) and (iii) by cross-multiplication, we have<\/p>\n

\\(a\\over 14 + 6\\) = \\(b\\over 15 – 7\\) = \\(c\\over -2 – 10\\) <\/p>\n

\\(\\implies\\) \\(a\\over 5\\) = \\(b\\over 2\\) = \\(c\\over -3\\) = \\(\\lambda\\) (say)<\/p>\n

\\(\\implies\\) a = 5 \\(\\lambda\\), b = 2 \\(\\lambda\\), c = -3 \\(\\lambda\\)<\/p>\n

Substituting the values of a, b and c in (i), we get<\/p>\n

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

\\(\\implies\\) 5(x – 2) + 2(y – 2) – 3(z + 1) = 0<\/p>\n

\\(\\implies\\) 5x + 2y – 3z = 17, which is the required equation of the plane.<\/p>\n\n\n

\n
Next – Intercept Form of a Plane \u2013 Equation & Example<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – General Equation of Plane 3D<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn the equation of plane passing through three points with example. Let’s begin – Equation of a Plane Passing Through a Given Point The general equation of a plane passing through a point \\((x_1, y_1, z_1\\)) is  \\(a(x – x_1) + b(y – y_1) + c(z – z_1)\\) = 0, where a, …<\/p>\n

Equation of Plane Passing Through Three Points<\/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":"\nEquation of Plane Passing Through Three Points - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn the equation of plane passing through three points with example.\" \/>\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-plane-passing-through-three-points\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Equation of Plane Passing Through Three Points - 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