{"id":6122,"date":"2021-10-07T18:13:59","date_gmt":"2021-10-07T12:43:59","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6122"},"modified":"2021-10-09T00:43:50","modified_gmt":"2021-10-08T19:13:50","slug":"distance-between-two-parallel-lines-in-3d","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/distance-between-two-parallel-lines-in-3d\/","title":{"rendered":"Distance Between Two Parallel Lines in 3d"},"content":{"rendered":"

Here you will learn formula to find the distance between two parallel lines in 3d with example.<\/p>\n

Let’s begin –<\/p>\n

Distance Between Two Parallel Lines in 3d<\/h2>\n

Let \\(l_1\\) and \\(l_2\\) be two parallel lines having vector equations<\/p>\n

\\(l_1\\) : \\(\\vec{r}\\) = \\(\\vec{a_1}\\) + \\(\\lambda\\)\\(\\vec{b}\\)<\/p>\n

and \\(l_2\\) : \\(\\vec{r}\\) = \\(\\vec{a_2}\\) + \\(\\mu\\)\\(\\vec{b}\\).<\/p>\n

The shortest distance (S.D.) between two the parallel lines \\(\\vec{r}\\) = \\(\\vec{a_1}\\) + \\(\\lambda\\)\\(\\vec{b}\\) and \\(\\vec{r}\\) = \\(\\vec{a_2}\\) + \\(\\mu\\)\\(\\vec{b}\\) is given by<\/p>\n

\n

d = |\\((\\vec{a_2} -\\vec{ a_1}) \\times \\vec{b}\\over |\\vec{b}|\\)|<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Find the shortest distance between the lines whose vector equations are<\/p>\n

\\(\\vec{r}\\) = (\\(\\hat{i} + 2\\hat{j} + 3\\hat{k}\\)) + \\(\\lambda\\)(\\(2\\hat{i} + 3\\hat{j} + 4\\hat{k}\\))<\/p>\n

and \\(\\vec{r}\\) = (\\(2\\hat{i} + 4\\hat{j} + 5\\hat{k}\\)) + \\(\\mu\\)(\\(4\\hat{i} + 6\\hat{j} + 8\\hat{k}\\))<\/p>\n

Solution<\/strong><\/span> : The vector equations of given lines are<\/p>\n

\\(\\vec{r}\\) = (\\(\\hat{i} + 2\\hat{j} + 3\\hat{k}\\)) + \\(\\lambda\\)(\\(2\\hat{i} + 3\\hat{j} + 4\\hat{k}\\))                            …………(i)<\/p>\n

and \\(\\vec{r}\\) = (\\(2\\hat{i} + 4\\hat{j} + 5\\hat{k}\\)) + 2\\(\\mu\\)(\\(2\\hat{i} + 3\\hat{j} + 4\\hat{k}\\))                         ………..(ii)<\/p>\n

Equation (ii) can be re-written as<\/p>\n

\\(\\vec{r}\\) = (\\(2\\hat{i} + 4\\hat{j} + 5\\hat{k}\\)) + \\({\\mu}’\\)(\\(2\\hat{i} + 3\\hat{j} + 4\\hat{k}\\))                             …………(iii)<\/p>\n

where  \\({\\mu}’\\) = 2\\(\\mu\\)<\/p>\n

These two lines passes through the points having position vectors \\(\\vec{a_1}\\) = \\(\\hat{i} + 2\\hat{j} + 3\\hat{k}\\) and \\(a_2\\) = \\(2\\hat{i} + 4\\hat{j} + 5\\hat{k}\\) respectively and both are parallel to the vector \\(\\vec{b}\\) = (\\(2\\hat{i} + 3\\hat{j} + 4\\hat{k}\\)).<\/p>\n

So, the shortest distance between them is given by<\/p>\n

S.D. =  |\\((\\vec{a_2} -\\vec{ a_1}) \\times \\vec{b}\\over |\\vec{b}|\\)|                                  ………(iv)<\/p>\n

Now, \\((\\vec{a_2} -\\vec{ a_1}) \\times \\vec{b}\\) = \\(\\hat{i} + 2\\hat{j} + 2\\hat{k}\\)  \\(\\times\\) (\\(2\\hat{i} + 3\\hat{j} + 4\\hat{k}\\))<\/p>\n

= \\(\\begin{vmatrix} \\hat{i} & \\hat{j} &  \\hat{k} \\\\  1 & 2 & 2 \\\\  2 & 3 & 4 \\end{vmatrix}\\) = \\(2\\hat{i} – 0\\hat{j} – k\\hat{k}\\)<\/p>\n

\\(\\therefore\\)  |\\((\\vec{a_2} -\\vec{ a_1}) \\times \\vec{b}\\)| = \\(\\sqrt{4 + 0 + 1}\\) = \\(\\sqrt{5}\\)<\/p>\n

and |\\(\\vec{b}\\)| = \\(\\sqrt{4 + 9 + 16}\\) = \\(\\sqrt{29}\\)<\/p>\n

Substituting the values of |\\((\\vec{a_2} -\\vec{ a_1}) \\times \\vec{b}\\) | and |\\(\\vec{b}\\)| in (iv), we obtain<\/p>\n

S.D. = \\(\\sqrt{5}\\over \\sqrt{29}\\)<\/p>\n\n\n

\n
Previous – Distance Between Two Lines in 3d<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn formula to find the distance between two parallel lines in 3d with example. Let’s begin – Distance Between Two Parallel Lines in 3d Let \\(l_1\\) and \\(l_2\\) be two parallel lines having vector equations \\(l_1\\) : \\(\\vec{r}\\) = \\(\\vec{a_1}\\) + \\(\\lambda\\)\\(\\vec{b}\\) and \\(l_2\\) : \\(\\vec{r}\\) = \\(\\vec{a_2}\\) + \\(\\mu\\)\\(\\vec{b}\\). The shortest …<\/p>\n

Distance Between Two Parallel Lines in 3d<\/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":"\nDistance Between Two Parallel Lines in 3d - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn formula to find the distance between two parallel lines in 3d 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\/distance-between-two-parallel-lines-in-3d\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Distance Between Two Parallel Lines in 3d - 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