{"id":6378,"date":"2021-10-13T18:13:41","date_gmt":"2021-10-13T12:43:41","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6378"},"modified":"2021-10-14T22:47:20","modified_gmt":"2021-10-14T17:17:20","slug":"equation-of-plane-passing-through-intersection-of-two-planes","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/equation-of-plane-passing-through-intersection-of-two-planes\/","title":{"rendered":"Equation of Plane Passing Through Intersection of Two Planes"},"content":{"rendered":"

Here you will learn what is the equation of plane passing through intersection of two planes with examples.<\/p>\n

Let’s begin –<\/p>\n

Equation of Plane Passing Through Intersection of Two Planes<\/h2>\n

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

The equation of a plane passing through the intersection of the planes \\(\\vec{r}.\\vec{n_1}\\) = \\(d_1\\)  and \\(\\vec{r}.\\vec{n_2}\\) = \\(d_2\\) is given by<\/p>\n

\n

(\\(\\vec{r}.\\vec{n_1}\\) – \\(d_1\\)) + \\(\\lambda\\)(\\(\\vec{r}.\\vec{n_2}\\) – \\(d_2\\)) = 0<\/p>\n

or,  \\(\\vec{r}\\).(\\(\\vec{n_1} + \\lambda\\vec{n_2}\\)) = \\(d_1 + \\lambda d_2\\),<\/p>\n

where \\(\\lambda\\) is an arbitrary constant.<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : Find the equation of the plane containing the line of intersection of the planes \\(\\vec{r}\\).(\\(\\hat{i} + 3\\hat{j} – \\hat{k}\\)) = 5 and \\(\\vec{r}\\).(\\(2\\hat{i} – \\hat{j} + \\hat{k}\\)) = 3 and passing through the point (2, 1, -2),<\/p>\n

Solution<\/span><\/strong> : The equation of the plane through the line of intersection of the given planes is<\/p>\n

[\\(\\vec{r}\\).(\\(\\hat{i} + 3\\hat{j} – \\hat{k}\\)) – 5] + \\(\\lambda\\)[\\(\\vec{r}\\).(\\(2\\hat{i} – \\hat{j} + \\hat{k}\\)) – 3] = 0   <\/p>\n

\\(\\implies\\) \\(\\vec{r}\\).[\\(1 + 2\\lambda)\\hat{i} + (3 – \\lambda)\\hat{j} + (-1 + \\lambda)\\hat{k}\\)] – 5 – 3\\(\\lambda\\) = 0                    ………..(i)<\/p>\n

If plane in (i) passes through (2, 1, -2), then the vector \\(2\\hat{i} + \\hat{j} – 2\\hat{k}\\) should satisfy it<\/p>\n

(\\(2\\hat{i} + \\hat{j} – 2\\hat{k}\\)).[\\(1 + 2\\lambda)\\hat{i} + (3 – \\lambda)\\hat{j} + (-1 + \\lambda)\\hat{k}\\)] – (5 + 3\\(\\lambda\\)) = 0<\/p>\n

\\(\\implies\\) \\(-2\\lambda + 2\\) = 0 \\(\\implies\\) \\(\\lambda\\) = 1 <\/p>\n

Putting \\(\\lambda\\) = 1 in equation (i), we obtain the equation of the required plane as<\/p>\n

\\(\\vec{r}\\).(\\(3\\hat{i} + 2\\hat{j} + 0\\hat{k}\\)) = 8.<\/p>\n

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

The equation of a plane passing through the intersection of planes \\(a_1x + b_1y + c_1z + d_1\\) = 0 and \\(a_2x + b_2y + c_2z + d_2\\) = 0 is<\/p>\n

\n

(\\(a_1x + b_1y + c_1z + d_1\\)) + \\(\\lambda\\)(\\(a_2x + b_2y + c_2z + d_2\\)) = 0,<\/p>\n

where \\(\\lambda\\) is a constant.<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : Find the equation of the plane containing the line of intersection of the planes x + y + z – 6 = 0 and 2x + 3y + 4z + 5 = 0 and passing through the point (1, 1, 1),<\/p>\n

Solution<\/span><\/strong> : The equation of the plane through the line of intersection of the given planes is<\/p>\n

(x + y + z – 6) + \\(\\lambda\\)(2x + 3y + 4z + 5) = 0                            ………..(i)<\/p>\n

If (i) passes through (1, 1, 1), then<\/p>\n

-3 + 14 \\(\\lambda\\) = 0 \\(\\implies\\) \\(\\lambda\\) = 3\/14<\/p>\n

Putting \\(\\lambda\\) = 3\/14 in equation (i), we obtain the equation of the required plane as<\/p>\n

(x + y + z – 6) + \\(3\\over 14\\) (2x + 3y + 4z + 5) = 0<\/p>\n

or, 20x + 23y + 26z – 69 = 0.<\/p>\n\n\n

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

Here you will learn what is the equation of plane passing through intersection of two planes with examples. Let’s begin – Equation of Plane Passing Through Intersection of Two Planes (a) Vector Form The equation of a plane passing through the intersection of the planes \\(\\vec{r}.\\vec{n_1}\\) = \\(d_1\\)  and \\(\\vec{r}.\\vec{n_2}\\) = \\(d_2\\) is given by …<\/p>\n

Equation of Plane Passing Through Intersection of Two Planes<\/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 Passing Through Intersection of Two Planes<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is the equation of plane passing through intersection of two planes 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