{"id":6388,"date":"2021-10-13T18:21:46","date_gmt":"2021-10-13T12:51:46","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6388"},"modified":"2021-10-14T22:49:59","modified_gmt":"2021-10-14T17:19:59","slug":"equation-of-plane-containing-two-lines","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/equation-of-plane-containing-two-lines\/","title":{"rendered":"Equation of Plane Containing Two Lines"},"content":{"rendered":"

Here you will learn how to find equation of plane containing two lines with examples.<\/p>\n

Let’s begin –<\/p>\n

Equation of Plane Containing Two Lines<\/h2>\n

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

If the lines \\(\\vec{r}\\) = \\(a_1 + \\lambda\\vec{b_1}\\) and \\(\\vec{r}\\) = \\(a_2 + \\mu\\vec{b_2}\\) are coplanar, then<\/p>\n

\n

\\(\\vec{r_1}\\).(\\(\\vec{b_1}\\times \\vec{b_2}\\)) = \\(\\vec{a_2}\\).(\\(\\vec{b_1}\\times \\vec{b_2}\\)) <\/p>\n

or,   [\\(\\vec{r}\\) \\(\\vec{b_1}\\) \\(\\vec{b_2}\\)] = [\\(\\vec{a_2}\\) \\(\\vec{b_1}\\) \\(\\vec{b_2}\\)]<\/p>\n<\/blockquote>\n

and the equation of the plane containing them is<\/p>\n

\n

\\(\\vec{r_1}\\).(\\(\\vec{b_1}\\times \\vec{b_2}\\)) = \\(\\vec{a_1}\\).(\\(\\vec{b_1}\\times \\vec{b_2}\\)) <\/p>\n

or,   \\(\\vec{r_1}\\).(\\(\\vec{b_1}\\times \\vec{b_2}\\)) = \\(\\vec{a_2}\\).(\\(\\vec{b_1}\\times \\vec{b_2}\\)) <\/p>\n<\/blockquote>\n

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

If the line \\(x – x_1\\over l_1\\) = \\(y – y_1\\over m_1\\) = \\(z – z_1\\over n_1\\) and \\(x – x_2\\over l_2\\) = \\(y – y_2\\over m_2\\) = \\(z – z_2\\over n_2\\) are coplanar then<\/p>\n

\n

\\(\\begin{vmatrix} x_2 – x_1 & y_2 – y_1 &  z_2 – z_1 \\\\  l_1 & m_1 & n_1 \\\\  l_2 & m_2 & n_2 \\end{vmatrix}\\) = 0<\/p>\n<\/blockquote>\n

and the equation of the plane containing them is<\/p>\n

\n

\\(\\begin{vmatrix} x – x_1 & y – y_1 &  z – z_1 \\\\  l_1 & m_1 & n_1 \\\\  l_2 & m_2 & n_2 \\end{vmatrix}\\) = 0 <\/p>\n

or,  \\(\\begin{vmatrix} x – x_2 & y – y_2 &  z – z_2 \\\\  l_1 & m_1 & n_1 \\\\  l_2 & m_2 & n_2 \\end{vmatrix}\\) = 0<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Prove that the lines \\(x + 1\\over 3\\) = \\(y + 3\\over 5\\) = \\(z + 5\\over 7\\) and \\(x – 2\\over 1\\) = \\(y – 4\\over 4\\) = \\(z – 6\\over 7\\) are coplanar. Also, find the plane containing these two lines.<\/p>\n

Solution<\/span><\/strong> : We know that the lines<\/p>\n

\\(x – x_1\\over l_1\\) = \\(y – y_1\\over m_1\\) = \\(z – z_1\\over n_1\\) and \\(x – x_2\\over l_2\\) = \\(y – y_2\\over m_2\\) = \\(z – z_2\\over n_2\\) are coplanar if<\/p>\n

\\(\\begin{vmatrix} x_2 – x_1 & y_2 – y_1 &  z_2 – z_1 \\\\  l_1 & m_1 & n_1 \\\\  l_2 & m_2 & n_2 \\end{vmatrix}\\) = 0<\/p>\n

and the equation of the plane containing these two lines is<\/p>\n

\\(\\begin{vmatrix} x – x_1 & y – y_1 &  z – z_1 \\\\  l_1 & m_1 & n_1 \\\\  l_2 & m_2 & n_2 \\end{vmatrix}\\) = 0<\/p>\n

Here, \\(x_1\\) = -1, \\(y_1\\) = -3, \\(z_1\\) = -5,  \\(x_2\\) = 2, \\(y_2\\) = 4, \\(z_2\\) = 6,<\/p>\n

 \\(l_1\\) = 3, \\(m_1\\) = 5, \\(n_1\\) = 7,  \\(l_2\\) = 1, \\(m_1\\) = 4, \\(n_1\\) = 7.<\/p>\n

\\(\\therefore\\) \\(\\begin{vmatrix} x_2 – x_1 & y_2 – y_1 &  z_2 – z_1 \\\\  l_1 & m_1 & n_1 \\\\  l_2 & m_2 & n_2 \\end{vmatrix}\\) = \\(\\begin{vmatrix} 3 & 7 &  11 \\\\  3 & 5 & 7 \\\\  1 & 4 & 7 \\end{vmatrix}\\) = 21 – 98 + 77 = 0<\/p>\n

So, the given lines are coplanar.<\/p>\n

The equation of the plane containing the lines is<\/p>\n

\\(\\begin{vmatrix} x + 1 & y + 3 &  z + 5 \\\\  3 & 5 & 7 \\\\  1 & 4 & 7 \\end{vmatrix}\\) = 0<\/p>\n

\\(\\implies\\) (x + 1)(35 – 28) – (y + 3)(21 – 7) + (z + 5)(12 – 5) = 0<\/p>\n

\\(\\implies\\) x – 2y + z = 0<\/p>\n\n\n

\n
Next – Image of a Point in a Plane \u2013 Formula & Example<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Intersection of a Line and a Plane<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn how to find equation of plane containing two lines with examples. Let’s begin – Equation of Plane Containing Two Lines (a) Vector Form If the lines \\(\\vec{r}\\) = \\(a_1 + \\lambda\\vec{b_1}\\) and \\(\\vec{r}\\) = \\(a_2 + \\mu\\vec{b_2}\\) are coplanar, then \\(\\vec{r_1}\\).(\\(\\vec{b_1}\\times \\vec{b_2}\\)) = \\(\\vec{a_2}\\).(\\(\\vec{b_1}\\times \\vec{b_2}\\))  or,   [\\(\\vec{r}\\) \\(\\vec{b_1}\\) \\(\\vec{b_2}\\)] = [\\(\\vec{a_2}\\) …<\/p>\n

Equation of Plane Containing Two Lines<\/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 Containing Two Lines - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to find equation of plane containing two lines 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\/equation-of-plane-containing-two-lines\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Equation of Plane Containing Two Lines - 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