{"id":6384,"date":"2021-10-13T18:18:42","date_gmt":"2021-10-13T12:48:42","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6384"},"modified":"2021-10-14T22:49:05","modified_gmt":"2021-10-14T17:19:05","slug":"angle-between-a-line-and-a-plane","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/angle-between-a-line-and-a-plane\/","title":{"rendered":"Angle Between a Line and a Plane"},"content":{"rendered":"

Here you will learn formula to find angle between a line and a plane with examples.<\/p>\n

Let’s begin –<\/p>\n

Angle Between a Line and a Plane<\/h2>\n

The angle between a line and a plane is the complement of the angle between the line and normal to the plane<\/p>\n

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

The angle \\(\\theta\\) between a lines \\(\\vec{r}\\) = \\(\\vec{a}\\) + \\(\\lambda\\vec{b}\\) and the plane \\(\\vec{r}\\).\\(\\vec{n}\\) = d is given by<\/p>\n

\\(sin\\theta\\) = \\(\\vec{b}.\\vec{n}\\over |\\vec{b}||\\vec{n}|\\).<\/p>\n<\/blockquote>\n

Condition of Perpendicularity<\/strong> : If the line is perpendicular to the plane, then it is parallel to the normal to the plane. Therefore, \\(\\vec{b}\\) and \\(\\vec{n}\\) are parallel.<\/p>\n

\n

\\(\\vec{b}\\times \\vec{n}\\)   or,  \\(\\vec{b}\\) = \\(\\lambda\\) \\(\\vec{n}\\) for some scalar \\(\\lambda\\)<\/p>\n<\/blockquote>\n

Condition of Parallelism<\/strong> : If the line is parallel to the plane, then it is perpendicular to the normal to the plane. Therefore, \\(\\vec{b}\\) and \\(\\vec{n}\\) are perpendicular<\/p>\n

\n

\\(\\vec{b}\\).\\(\\vec{n}\\) = 0<\/p>\n<\/blockquote>\n

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

The angle \\(\\theta\\) between the lines \\(x – x1\\over l\\) = \\(y – y1\\over m\\) = \\(z – z1\\over n\\) and the plane ax + by + cz + d = 0 is given by<\/p>\n

\\(sin\\theta\\) = \\(al + bm + cn\\over \\sqrt{a^2 + b^2 + c^2}\\sqrt{l^2 + m^2 + n^2}\\)<\/p>\n<\/blockquote>\n

Condition of Perpendicularity<\/strong> : If the line is perpendicular to the plane, then it is parallel to its normal. Therefore,<\/p>\n

\n

\\(l\\over a\\) = \\(m\\over b\\) = \\(n\\over c\\)<\/p>\n<\/blockquote>\n

Condition of Parallelism<\/strong> : If the lines is parallel to the plane, then it is perpendicular to its normal. Therefore,<\/p>\n

\n

\\(\\vec{b}\\).\\(\\vec{n}\\) = 0 \\(\\implies\\)  al + bm + cn = 0<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Find the angle between the line \\(x + 1\\over 3\\) = \\(y – 1\\over 2\\) = \\(z – 2\\over 4\\) and the plane 2x + y – 3z + 4 = 0.<\/p>\n

Solution<\/span><\/strong> : Here, l = 3, m = 2 and  = 4<\/p>\n

and, a = 2, b = 1 and c = -3<\/p>\n

So, Angle between them is \\(sin\\theta\\) = \\(al + bm + cn\\over \\sqrt{a^2 + b^2 + c^2}\\sqrt{l^2 + m^2 + n^2}\\)<\/p>\n

= \\(6 + 3 – 12\\over \\sqrt{4 + 1 + 9}\\sqrt{9 + 4 + 16}\\) = \\(-4\\over \\sqrt{406}\\)<\/p>\n

\\(\\implies\\) \\(\\theta\\) = \\(sin^{-1}({-4\\over \\sqrt{406}})\\)<\/p>\n\n\n

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

Here you will learn formula to find angle between a line and a plane with examples. Let’s begin – Angle Between a Line and a Plane The angle between a line and a plane is the complement of the angle between the line and normal to the plane (a) Vector Form The angle \\(\\theta\\) between …<\/p>\n

Angle Between a Line and 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":"\nAngle Between a Line and a Plane - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn formula to find angle between a line and a 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\/angle-between-a-line-and-a-plane\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Angle Between a Line and a Plane - 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