{"id":6376,"date":"2021-10-13T18:09:56","date_gmt":"2021-10-13T12:39:56","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6376"},"modified":"2021-10-14T22:46:57","modified_gmt":"2021-10-14T17:16:57","slug":"equation-of-plane-parallel-to-plane","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/equation-of-plane-parallel-to-plane\/","title":{"rendered":"Equation of Plane Parallel to Plane"},"content":{"rendered":"

Here you will learn equation of plane parallel to plane with examples.<\/p>\n

Let’s begin – <\/p>\n

Equation of Plane Parallel to Plane<\/h2>\n

(a) Vector Form<\/h3>\n

Since parallel planes have the common normal, therefore equation of a plane parallel to the plane \\(\\vec{r}\\).\\(\\vec{n}\\) = \\(\\vec{d_1}\\) is<\/p>\n

\n

\\(\\vec{r}\\).\\(\\vec{n}\\) = \\(\\vec{d_2}\\)<\/p>\n<\/blockquote>\n

where \\(\\vec{d_2}\\) is constant determined by the given condition.<\/p>\n

Example<\/strong><\/span> : Find the equation of plane passing through the point \\(\\hat{i} + \\hat{j} + \\hat{k}\\) and parallel to the plane \\(\\vec{r}\\).\\(2\\hat{i} – \\hat{j} + 2\\hat{k}\\) = 5.<\/p>\n

Solution<\/span><\/strong> : The equation of a plane parallal to the plane \\(\\vec{r}\\).(\\(2\\hat{i} – \\hat{j} + 2\\hat{k}\\)) = 5 is<\/p>\n

\\(\\vec{r}\\).(\\(2\\hat{i} – \\hat{j} + 2\\hat{k}\\)) = d                        ………(i)<\/p>\n

If it passes through \\(\\hat{i} + \\hat{j} + \\hat{k}\\), then<\/p>\n

(\\(\\hat{i} + \\hat{j} + \\hat{k}\\)).(\\(2\\hat{i} – \\hat{j} + 2\\hat{k}\\)) = d \\(\\implies\\) 2 – 1 + 2 = d \\(\\implies\\) d = 3<\/p>\n

Putting d = 3 in (i), we obtain \\(\\vec{r}\\).(\\(2\\hat{i} – \\hat{j} + 2\\hat{k}\\)) = 3 as the equation of the required plane.<\/p>\n

(b) Cartesian Form<\/h3>\n

Let ax + by + cz + d = 0 be the cartesian equation of a plane. Then, direction ratios of its normal are proportional to a, b, c.<\/p>\n

Since parallel planes have common normals.<\/p>\n

Therefore, the direction ratios of the normal to the parallel plane are also proportional to a, b, c.<\/p>\n

Thus, the equation of plane parallal to the plane ax + by + cz + d = 0 is <\/p>\n

\n

ax + by + cz + k = 0,<\/p>\n<\/blockquote>\n

where k is an arbitrary constant and is determined by the given condition.<\/p>\n

Example<\/strong><\/span> : Find the equation of plane passing through the point (1, 4, -2) and parallal to the plane -2x + y – 3z = 7.<\/p>\n

Solution<\/span><\/strong> : Let the equation of a plane parallal to the plane -2x + y – 3z = 7 be<\/p>\n

-2x + y – 3z + k = 7                      ………(i)<\/p>\n

If it passes through (1, 4, -2), then<\/p>\n

(-2)(1) + 4 – 3(-2) + k = 0 \\(\\implies\\) k = -8<\/p>\n

Putting k = -8 in (i), we obtain -2x + y – 3z – 8 = 0 as the equation of the required plane.<\/p>\n\n\n

\n
Next – Equation of Plane Passing Through Intersection of Two Planes<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Angle Between Two Planes Formula<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn equation of plane parallel to plane with examples. Let’s begin –  Equation of Plane Parallel to Plane (a) Vector Form Since parallel planes have the common normal, therefore equation of a plane parallel to the plane \\(\\vec{r}\\).\\(\\vec{n}\\) = \\(\\vec{d_1}\\) is \\(\\vec{r}\\).\\(\\vec{n}\\) = \\(\\vec{d_2}\\) where \\(\\vec{d_2}\\) is constant determined by the given …<\/p>\n

Equation of Plane Parallel to 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":"\nEquation of Plane Parallel to Plane - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn equation of plane parallel to 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\/equation-of-plane-parallel-to-plane\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Equation of Plane Parallel to Plane - 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