{"id":6374,"date":"2021-10-13T18:07:36","date_gmt":"2021-10-13T12:37:36","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6374"},"modified":"2021-10-14T22:46:15","modified_gmt":"2021-10-14T17:16:15","slug":"angle-between-two-planes-formula","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/angle-between-two-planes-formula\/","title":{"rendered":"Angle Between Two Planes Formula"},"content":{"rendered":"

Here you will learn how to find angle between two planes formula with examples.<\/p>\n

Let’s begin –<\/p>\n

Angle Between Two Planes Formula<\/h2>\n

The angle between two planes is defined as the angle between their normals.<\/p>\n

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

The angle \\(\\theta\\) between the planes \\(\\vec{r}\\).\\(\\vec{n_1}\\) = \\(\\vec{d_1}\\) and \\(\\vec{r}\\).\\(\\vec{n_2}\\) = \\(\\vec{d_2}\\) is given by<\/p>\n

\\(cos\\theta\\) = \\(\\vec{n_1}.\\vec{n_2}\\over |\\vec{n_1}||\\vec{n_1}|\\).<\/p>\n<\/blockquote>\n

Condition of Perpendicularity<\/strong> : If the planes \\(\\vec{r}\\).\\(\\vec{n_1}\\) = \\(\\vec{d_1}\\) and \\(\\vec{r}\\).\\(\\vec{n_2}\\) = \\(\\vec{d_2}\\) are perpendicular, then \\(\\vec{n_1}\\) and \\(\\vec{n_2}\\) are perpendicular.<\/p>\n

\n

\\(\\vec{n_1}\\).\\(\\vec{n_2}\\) = 0<\/p>\n<\/blockquote>\n

Condition of Parallelism<\/strong> : If the planes \\(\\vec{r}\\).\\(\\vec{n_1}\\) = \\(\\vec{d_1}\\) and \\(\\vec{r}\\).\\(\\vec{n_2}\\) = \\(\\vec{d_2}\\) are parallel, then \\(\\vec{n_1}\\) and \\(\\vec{n_2}\\) are parallel.<\/p>\n

\n

Therefore, there exist a scalar \\(\\lambda\\) such that \\(\\vec{n_1}\\) = \\(\\lambda\\) \\(\\vec{n_2}\\)<\/p>\n<\/blockquote>\n

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

The angle \\(\\theta\\) between the planes \\(a_1x + b_1y + c_1z + d_1\\) = 0 and \\(a_2x + b_2y + c_2z + d_2\\) = 0 is given by<\/p>\n

\\(cos\\theta\\) = \\(a_1a_2 + b_1b_2 + c_1c_2\\over \\sqrt{{a_1}^2 + {b_1}^2 + {c_1}^2}\\sqrt{{a_1}^2 + {b_1}^2 + {c_1}^2}\\)<\/p>\n<\/blockquote>\n

Condition of Perpendicularity<\/strong> : If the planes are perpendicular. Then \\(\\vec{n_1}\\) and \\(\\vec{n_2}\\) are perpendicular<\/p>\n

\n

\\(a_1a_2 + b_1b_2 + c_1c_2\\) = 0<\/p>\n<\/blockquote>\n

Condition of Parallelism<\/strong> : If the lines are parallel, then \\(\\vec{n_1}\\) and \\(\\vec{n_2}\\) are parallel, <\/p>\n

\n

\\(a_1\\over a_2\\) = \\(b_1\\over b_2\\) = \\(c_1\\over c_2\\)<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Find the angle between the planes \\(\\vec{r}\\).(\\(2\\hat{i} – \\hat{j} + \\hat{k}\\)) = 6 and \\(\\vec{r}\\).(\\(\\hat{i} + \\hat{j} + 2\\hat{k}\\)) = 5.<\/p>\n

Solution<\/span><\/strong> : We know that the angle between the planes \\(\\vec{r}\\).\\(\\vec{n_1}\\) = \\(\\vec{d_1}\\) and \\(\\vec{r}\\).\\(\\vec{n_2}\\) = \\(\\vec{d_2}\\) is given by<\/p>\n

\\(cos\\theta\\) = \\(\\vec{n_1}.\\vec{n_2}\\over |\\vec{n_1}||\\vec{n_1}|\\)<\/p>\n

Here \\(n_1\\) = \\(2\\hat{i} – \\hat{j} + \\hat{k}\\) and \\(n_2\\) = \\(\\hat{i} + \\hat{j} + 2\\hat{k}\\)<\/p>\n

\\(\\therefore\\)  \\(cos\\theta\\) = \\((2\\hat{i} – \\hat{j} + \\hat{k}).(\\hat{i} + \\hat{j} + 2\\hat{k})\\over |2\\hat{i} – \\hat{j} + \\hat{k}||\\hat{i} + \\hat{j} + 2\\hat{k}|\\)<\/p>\n

= \\(2 – 1 + 2\\over \\sqrt{4 + 1 + 1} \\sqrt{1 + 1 +4}\\) = \\(1\\over 2\\)<\/p>\n

\\(\\implies\\) \\(\\theta\\) = \\(\\pi\\over 3\\)<\/p>\n\n\n

\n
Next – Equation of Plane Parallel to Plane<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Equation of Plane in Normal Form<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn how to find angle between two planes formula with examples. Let’s begin – Angle Between Two Planes Formula The angle between two planes is defined as the angle between their normals. (a) Vector Form The angle \\(\\theta\\) between the planes \\(\\vec{r}\\).\\(\\vec{n_1}\\) = \\(\\vec{d_1}\\) and \\(\\vec{r}\\).\\(\\vec{n_2}\\) = \\(\\vec{d_2}\\) is given by \\(cos\\theta\\) …<\/p>\n

Angle Between Two Planes Formula<\/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 Two Planes Formula - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to find angle between two planes formula 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-two-planes-formula\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Angle Between Two Planes Formula - 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