{"id":6380,"date":"2021-10-13T18:15:24","date_gmt":"2021-10-13T12:45:24","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6380"},"modified":"2021-10-14T22:47:57","modified_gmt":"2021-10-14T17:17:57","slug":"distance-of-a-point-from-a-plane","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/distance-of-a-point-from-a-plane\/","title":{"rendered":"Distance of a Point From a Plane"},"content":{"rendered":"

Here you will learn how to find the distance of a point from a plane formula with examples.<\/p>\n

Let’s begin –<\/p>\n

Distance of a Point from a Plane<\/h2>\n

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

The length of the perpendicular from a point having position vector \\(\\vec{a}\\) to the plane \\(\\vec{r}\\).\\(\\vec{n}\\) = d  is<\/p>\n

\n

p = \\(|\\vec{a}.\\vec{n} – d|\\over |\\vec{n}|\\)<\/p>\n<\/blockquote>\n

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

The length of the perpendicular form a point P\\((x_1, y_1, z_1)\\) to the plane ax + by + cz + d = 0 is given by<\/p>\n

\n

p = \\(|ax_1 + by_1 + cz_1 + d|\\over \\sqrt{a^2 + b^2 + c^2}\\)<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Find the distance of the point \\(2\\hat{i} + \\hat{j} – \\hat{k}\\) from the plane \\(\\vec{r}\\).\\(\\hat{i}  – 2\\hat{j} + 4\\hat{k}\\) = 9.<\/p>\n

Solution<\/strong><\/span> : We know that the perpendicular distance of a point with the position vector \\(\\vec{a}\\) from the given planes \\(\\vec{r}\\).\\(\\vec{n}\\) = d is given by<\/p>\n

p = \\(|\\vec{a}.\\vec{n} – d|\\over |\\vec{n}|\\)<\/p>\n

Here, \\(\\vec{a}\\) = \\(2\\hat{i} + \\hat{j} – \\hat{k}\\)  ,  \\(\\vec{n}\\) = \\(\\hat{i}  – 2\\hat{j} + 4\\hat{k}\\) and d = 9.<\/p>\n

So, the required distance p is given by<\/p>\n

p = \\(|(2\\hat{i} + \\hat{j} – \\hat{k}).(\\hat{i}  – 2\\hat{j} + 4\\hat{k}) – 9|\\over \\sqrt{1 + 4 + 16}\\)<\/p>\n

= \\(|2 – 2 – 4 – 9|\\over \\sqrt{21}\\) = \\(13\\over \\sqrt{21}\\)<\/p>\n

Example<\/strong><\/span> : Find the distance of the point (2, 1, 0) from the plane 2x + y + 2z + 5 = 0.<\/p>\n

Solution<\/strong><\/span> : We know that the perpendicular distance of the point \\((x_1, y_1, z_1)\\) from the given planes ax+ by + cz + d = 0 is given by<\/p>\n

p = \\(|ax_1 + by_1 + cz_1 + d|\\over \\sqrt{a^2 + b^2 + c^2}\\)<\/p>\n

So, the required distance p is<\/p>\n

p = \\(|2\\times 2 + 1 + 2\\times 0 + 5|\\over \\sqrt{4 + 1 + 4}\\) = \\(10\\over 3\\)<\/p>\n\n\n

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

Here you will learn how to find the distance of a point from a plane formula with examples. Let’s begin – Distance of a Point from a Plane (a) Vector Form The length of the perpendicular from a point having position vector \\(\\vec{a}\\) to the plane \\(\\vec{r}\\).\\(\\vec{n}\\) = d  is p = \\(|\\vec{a}.\\vec{n} – d|\\over …<\/p>\n

Distance of a Point From 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":"\nDistance of a Point From a Plane - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to find the distance of a point from a plane formula in both vector and cartesian form 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\/distance-of-a-point-from-a-plane\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Distance of a Point From a Plane - 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