{"id":6104,"date":"2021-10-07T17:32:17","date_gmt":"2021-10-07T12:02:17","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6104"},"modified":"2021-10-08T01:35:16","modified_gmt":"2021-10-07T20:05:16","slug":"direction-cosines-and-direction-ratios-of-line","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/direction-cosines-and-direction-ratios-of-line\/","title":{"rendered":"Direction Cosines and Direction Ratios of Line"},"content":{"rendered":"

In this post you will learn how to find direction cosines and direction ratios of line of the vector with examples.<\/p>\n

Let’s begin –<\/p>\n

Direction Cosines and Direction Ratios of Line<\/h2>\n

Direction cosines<\/strong><\/h4>\n
\n

The direction cosines of a line are defined as the direction cosines of any vector whose support is the given line.<\/p>\n<\/blockquote>\n

It follows from the above definition if A and B are two points on a given line L, then the direction cosines of vectors \\(\\vec{AB}\\) or, \\(\\vec{BA}\\) are the direction cosines of line L. Thus, if \\(\\alpha\\), \\(\\beta\\), \\(\\gamma\\) are the angles which the line L makes with the positive direction of x-axis, y-axis and z-axis respectively, then its direction cosines are either, \\(cos\\alpha\\), \\(cos\\beta\\), \\(cos\\gamma\\) or – \\(cos\\alpha\\), – \\(cos\\beta\\), – \\(cos\\gamma\\).<\/p>\n

Therefore, if l, m, n are direction cosines of a line, then -l, -m, -n are also its direction cosines and we always have<\/p>\n

\n

\\(l^2 + m^2 + n^2\\) = 1<\/p>\n<\/blockquote>\n

If A\\((x_1, y_1, z_1)\\) and B\\((x_2, y_2, z_2)\\) are two points on a line L, then its direction cosines are<\/p>\n

\n

\\(x_2 – x_1\\over AB\\), \\(y_2 – y_1\\over AB\\), \\(z_2 – z_1\\over AB\\) or \\(x_1 – x_2\\over AB\\), \\(y_1 – y_2\\over AB\\), \\(z_1 – z_2\\over AB\\)<\/p>\n<\/blockquote>\n

Direction Ratios<\/strong><\/h4>\n
\n

The direction ratios of a line are proportional to the direction ratios of any vector whose support is the given line.<\/p>\n<\/blockquote>\n

If A\\((x_1, y_1, z_1)\\) and B\\((x_2, y_2, z_2)\\) are two points on a line L, then its direction ratios are proportional to<\/p>\n

\n

\\(x_2 – x_1\\), \\(y_2 – y_1\\), \\(z_2 – z_1\\)<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Find the direction cosines and direction ratios of the line whose end points are A(1, 2, 3) and B(5, 8, 11).<\/p>\n

Solution<\/span><\/strong> : We have, A(1, 2, 3) and B(5, 8, 11)<\/p>\n

Direction ratios<\/strong> = (4, 6, 8)<\/p>\n

AB = \\(\\sqrt{16 + 36 + 64}\\) = \\(\\sqrt{116}\\)<\/p>\n

Direction Cosines<\/strong> = (\\(4\\over \\sqrt{116}\\), \\(6\\over \\sqrt{116}\\), \\(8\\over \\sqrt{116}\\)).<\/p>\n\n\n

\n
Next – Formula for Angle Between Two Vectors<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Section Formula in 3d<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

In this post you will learn how to find direction cosines and direction ratios of line of the vector with examples. Let’s begin – Direction Cosines and Direction Ratios of Line Direction cosines The direction cosines of a line are defined as the direction cosines of any vector whose support is the given line. It …<\/p>\n

Direction Cosines and Direction Ratios of Line<\/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":"\nDirection Cosines and Direction Ratios of Line - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to find direction cosines and direction ratios of line 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\/direction-cosines-and-direction-ratios-of-line\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Direction Cosines and Direction Ratios of Line - 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