{"id":6372,"date":"2021-10-13T18:03:28","date_gmt":"2021-10-13T12:33:28","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6372"},"modified":"2021-10-14T22:45:49","modified_gmt":"2021-10-14T17:15:49","slug":"equation-of-plane-in-normal-form","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/equation-of-plane-in-normal-form\/","title":{"rendered":"Equation of Plane in Normal Form"},"content":{"rendered":"

Here you will learn equation of plane in normal form with example.<\/p>\n

Let’s begin –<\/p>\n

Equation of Plane in Normal Form<\/h2>\n

(a) Vector Form <\/h3>\n
\n

The vector equation of a plane normal to unit vector \\(\\hat{n}\\) and at a distance d from the origin is<\/p>\n

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

Remark 1 :<\/strong> The vector equation of ON is \\(\\vec{r}\\) = \\(\\vec{0}\\) + \\(\\lambda\\) \\(\\hat{n}\\) and the position vector of N is d\\(\\hat{n}\\) as it is at a distance d from the origin from the origin O.<\/p>\n

(b) Cartesian Form <\/h3>\n
\n

If l, m, n are direction cosines of the normal to a given plane which is at a distance p from the origin, then the equation of the plane is<\/p>\n

lx + my + nz = p<\/p>\n<\/blockquote>\n

Note<\/strong> : The equation \\(\\vec{r}.\\vec{n}\\) = d is in normal form if \\(\\vec{n}\\) is a unit vector and in such a case d on the right hand side denotes the distance of the plane from the origin. If \\(\\vec{n}\\) is not a unit vector, then to reduce the equation \\(\\vec{r}.\\vec{n}\\) = d to normal form divide both sides by | \\(\\vec{n}\\) | to obtain<\/p>\n

\\(\\vec{r}\\).\\(\\vec{n}\\over |\\vec{n}|\\) = \\(d\\over |\\vec{n}|\\) \\(\\implies\\) \\(\\vec{r}.\\hat{n}\\) = \\(d\\over |\\vec{n}|\\)<\/p>\n

Example<\/strong><\/span> : Find the vector equation of a plane which is at a distance of 8 units from the origin and which is normal to the vector \\(2\\hat{i} + \\hat{j} + 2\\hat{k}\\).<\/p>\n

Solution<\/span><\/strong> : Here, d = 8 and \\(\\vec{n}\\) = \\(2\\hat{i} + \\hat{j} + 2\\hat{k}\\)<\/p>\n

\\(\\therefore\\) \\(\\hat{n}\\) = \\(\\vec{n}\\over |\\vec{n}|\\) = \\(2\\hat{i} + \\hat{j} + 2\\hat{k}\\over \\sqrt{4 + 1 + 4}\\)<\/p>\n

= \\({2\\over 3}\\hat{i} + {1\\over 3}\\hat{j} + {2\\over 3}\\hat{k}\\)<\/p>\n

Hence, the required equation of the plane is<\/p>\n

\\(\\vec{r}\\).(\\({2\\over 3}\\hat{i} + {1\\over 3}\\hat{j} + {2\\over 3}\\hat{k}\\)) = 8                    <\/p>\n

[ By using \\(\\vec{r}.\\hat{n}\\) = d ]<\/p>\n

or, \\(\\vec{r}\\).(\\(2\\hat{i} + \\hat{j} + 2\\hat{k}\\)) = 24<\/p>\n\n\n

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

Here you will learn equation of plane in normal form with example. Let’s begin – Equation of Plane in Normal Form (a) Vector Form  The vector equation of a plane normal to unit vector \\(\\hat{n}\\) and at a distance d from the origin is \\(\\vec{r}.\\hat{n}\\) = d Remark 1 : The vector equation of ON …<\/p>\n

Equation of Plane in Normal Form<\/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 in Normal Form - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn equation of plane in normal form i.e. in both vector and cartesian form 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-in-normal-form\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Equation of Plane in Normal Form - 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