{"id":6108,"date":"2021-10-07T17:41:08","date_gmt":"2021-10-07T12:11:08","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6108"},"modified":"2021-10-09T00:40:03","modified_gmt":"2021-10-08T19:10:03","slug":"equation-of-a-line-in-vector-form","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/equation-of-a-line-in-vector-form\/","title":{"rendered":"Equation of a Line in Vector Form"},"content":{"rendered":"

Here you will learn equation of a line in vector form passing through a fixed point and passing through two points.<\/p>\n

Let’s begin –<\/p>\n

Equation of a Line in Vector Form<\/h2>\n

The vector equation of a straight line passing through a fixed point with position vector \\(\\vec{a}\\) and parallel to a given vector \\(\\vec{b}\\) is <\/p>\n

\n

\\(\\vec{r}\\) = \\(\\vec{a}\\) + \\(\\lambda \\vec{b}\\), where \\(\\lambda\\) is scalar.<\/p>\n<\/blockquote>\n

Note<\/b> : In the above equation \\(\\vec{r}\\) is the position vector of any point P (x, y, z) on the line. Therefore, \\(\\vec{r}\\) = \\(x\\hat{i} + y\\hat{j} + z\\hat{k}\\).<\/p>\n

Example<\/strong><\/span> : Find the vector equation of a line which passes through the point with position vector \\(2\\hat{i} – \\hat{j} + 4\\hat{k}\\) and is in the direction \\(\\hat{i} + \\hat{j} – 2\\hat{k}\\).<\/p>\n

Solution<\/span><\/strong> : Here \\(\\vec{a}\\) = \\(2\\hat{i} – \\hat{j} + 4\\hat{k}\\) and \\(\\vec{b}\\) = \\(\\hat{i} + \\hat{j} – 2\\hat{k}\\).<\/p>\n

So, the vector equation of the required line is<\/p>\n

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

or, \\(\\vec{r}\\) = (\\(2\\hat{i} – \\hat{j} + 4\\hat{k}\\)) + \\(\\lambda (\\hat{i} + \\hat{j} – 2\\hat{k})\\), where \\(\\lambda\\) is a scalar.<\/p>\n

Equation of Line in Vector Form Passing Through Two Points<\/strong><\/h4>\n

The vector equation of line passing through two points with position vectors \\(\\vec{a}\\) and \\(\\vec{b}\\) is<\/p>\n

\n

\\(\\vec{r}\\) = \\(\\lambda\\) \\((\\vec{b} – \\vec{a})\\), where \\(\\lambda\\) is a scalar<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Find the vector equation of a line which passes through the point A (3, 4, -7) and B (1, -1, 6)<\/p>\n

Solution<\/span><\/strong> : We know that the vector equation of line passing through two points with position vectors \\(\\vec{a}\\) and \\(\\vec{b}\\) is,<\/p>\n

\\(\\vec{r}\\) = \\(\\lambda\\) \\((\\vec{b} – \\vec{a})\\)<\/span><\/p>\n

Here \\(\\vec{a}\\) = \\(3\\hat{i} + 4\\hat{j} – 7\\hat{k}\\) and \\(\\vec{b}\\) = \\(\\hat{i} – \\hat{j} + 6\\hat{k}\\).<\/p>\n

So, the vector equation of the required line is<\/p>\n

 \\(\\vec{r}\\) = (\\(3\\hat{i} + 4\\hat{j} – 7\\hat{k}\\)) + \\(\\lambda\\)  (\\(\\hat{i} – \\hat{j} + 6\\hat{k}\\) – \\(3\\hat{i} + 4\\hat{j} – 7\\hat{k}\\))<\/p>\n

or, \\(\\vec{r}\\) = (\\(3\\hat{i} + 4\\hat{j} – 7\\hat{k}\\)) + \\(\\lambda\\) (\\(-2\\hat{i} – 5\\hat{j} + 13\\hat{k}\\))<\/p>\n

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

\n
Next – Cartesian Equation of a Line<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Formula for Angle Between Two Vectors<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn equation of a line in vector form passing through a fixed point and passing through two points. Let’s begin – Equation of a Line in Vector Form The vector equation of a straight line passing through a fixed point with position vector \\(\\vec{a}\\) and parallel to a given vector \\(\\vec{b}\\) is  …<\/p>\n

Equation of a Line in Vector Form<\/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 a Line in Vector Form - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn equation of a line in vector form in 3d passing through a fixed point and passing through two points.\" \/>\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-a-line-in-vector-form\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Equation of a Line in Vector Form - 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