{"id":6118,"date":"2021-10-07T18:10:39","date_gmt":"2021-10-07T12:40:39","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6118"},"modified":"2021-10-09T00:45:09","modified_gmt":"2021-10-08T19:15:09","slug":"perpendicular-distance-of-a-point-from-a-line-in-3d","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/perpendicular-distance-of-a-point-from-a-line-in-3d\/","title":{"rendered":"Perpendicular Distance of a Point From a Line in 3d"},"content":{"rendered":"

Here you will learn how to find perpendicular distance of a point from a line in 3d in both vector form and cartesian form.<\/p>\n

Let’s begin –<\/p>\n

Perpendicular Distance of a Point From a Line in 3d<\/h2>\n

(a) Cartesian Form<\/strong><\/h4>\n

Algorithm :<\/strong><\/p>\n

\n

Let P\\((\\alpha, \\beta, \\gamma)\\) be the given point, and let the given line be<\/p>\n

\\(x – x_1\\over a\\) = \\(y – y_1\\over b\\) = \\(z – z_1\\over c\\)<\/p>\n

1).<\/strong> Write the coordinates of a general point on the given line. The coordinates of general point on the line are (\\(x_1 + a\\lambda\\), \\(y_1 + b\\lambda\\), \\(z_1 + c\\lambda\\)), where \\(\\lambda\\) is a parameter. Assume that this point L is the foot of the perpendicular drawn from P on the given line.<\/p>\n

2).<\/strong> Write direction ratios of PL.<\/p>\n

3).<\/strong> Apply the condition of perpedidularity of the given line and PL.<\/p>\n

4).<\/strong> Obtain the value of \\(\\lambda\\)  from step 3.<\/p>\n

5).<\/strong> Substitute \\(\\lambda\\) in (\\(x_1 + a\\lambda\\), \\(y_1 + b\\lambda\\), \\(z_1 + c\\lambda\\)) to obtain the coordinates of L.<\/p>\n

6).<\/strong> Obtain PL by using distance formula.<\/p>\n<\/blockquote>\n

(b) Vector Form<\/strong><\/h4>\n

Algorithm :<\/strong><\/p>\n

\n

Let P(\\(\\vec{\\alpha})\\) be the given point, and let \\(\\vec{r}\\) = \\(\\vec{a}\\) + \\(\\lambda \\vec{b}\\) be the given line be<\/p>\n

1).<\/strong> Write the position vector of a general point on the given line. The position vector of a general point on \\(\\vec{r}\\) = \\(\\vec{a}\\) + \\(\\lambda \\vec{b}\\)  is \\(\\vec{a}\\) + \\(\\lambda \\vec{b}\\), where \\(\\lambda\\) is a parameter. Assume that this point L is required foot of the perpendicular from P on the given line.<\/p>\n

2).<\/strong> Obtain \\(\\vec{PL}\\) = Position vector of L – Position Vector of P = \\(\\vec{a}\\) + \\(\\lambda \\vec{b}\\) – \\(\\vec{\\alpha}\\).<\/p>\n

3).<\/strong> Put \\(\\vec{PL}\\).\\(\\vec{b}\\) = 0 i.e. (\\(\\vec{a}\\) + \\(\\lambda \\vec{b}\\) – \\(\\vec{\\alpha}\\)).\\(\\vec{b}\\) = 0 to obtain the value of \\(\\lambda\\).<\/p>\n

4).<\/strong> Substitute the value of \\(\\lambda\\) in \\(\\vec{r}\\) = \\(\\vec{a}\\) + \\(\\lambda \\vec{b}\\) to obtain the position vector of L.<\/p>\n

5).<\/strong> Find |\\(\\vec{PL}\\)| to obtain the required length of the perpendicular.<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Find the foot of the perpendicular from the point (0, 2, 3) on the line \\(x + 3\\over 5\\) = \\(y – 1\\over 2\\) = \\(z + 4\\over 3\\).<\/p>\n

Solution<\/span><\/strong> : Let L be the foot of the perpendicular drawn from the point P (0, 2, 3) to the given line.<\/p>\n

The coordinates of a general point on the line \\(x + 3\\over 5\\) = \\(y – 1\\over 2\\) = \\(z + 4\\over 3\\) is given by<\/p>\n

\\(x + 3\\over 5\\) = \\(y – 1\\over 2\\) = \\(z + 4\\over 3\\) = \\(\\lambda\\)<\/p>\n

or, x = \\(5\\lambda – 3\\), y = \\(2\\lambda + 1\\), z = \\(3\\lambda – 4\\).<\/p>\n

Let the coordinates of L be (\\(5\\lambda – 3\\), \\(2\\lambda + 1\\), \\(3\\lambda – 4\\)). Therefore, direction ratios of PL are proportional to<\/p>\n

\\(5\\lambda – 3\\) – 0, \\(2\\lambda + 1\\) – 2, \\(3\\lambda – 4\\) – 3 i.e. \\(5\\lambda – 3\\), \\(2\\lambda – 1\\), \\(3\\lambda – 7\\).<\/p>\n

Direction ratios of the given line are proportional to 5, 2, 3.<\/p>\n

But, PL is perpendicular to the given line.<\/p>\n

\\(\\therefore\\)  5(\\(5\\lambda – 3\\)) + 2(\\(2\\lambda – 1\\)) + 3(\\(3\\lambda – 7\\)) = 0 \\(\\implies\\) \\(\\lambda\\) = 1<\/p>\n

Putting \\(\\lambda\\) = 1 in (\\(5\\lambda – 3\\), \\(2\\lambda + 1\\), \\(3\\lambda – 4\\)), the coordinates of L are (2, 3, -1).<\/p>\n\n\n

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

Here you will learn how to find perpendicular distance of a point from a line in 3d in both vector form and cartesian form. Let’s begin – Perpendicular Distance of a Point From a Line in 3d (a) Cartesian Form Algorithm : Let P\\((\\alpha, \\beta, \\gamma)\\) be the given point, and let the given line …<\/p>\n

Perpendicular Distance of a Point From a Line 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":"\nPerpendicular Distance of a Point From a Line in 3d - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to find perpendicular distance of a point from a line in 3d in both vector form and cartesian form.\" \/>\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\/perpendicular-distance-of-a-point-from-a-line-in-3d\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Perpendicular Distance of a Point From a Line in 3d - 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