{"id":3915,"date":"2021-08-10T03:13:31","date_gmt":"2021-08-10T03:13:31","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3915"},"modified":"2021-10-10T19:02:07","modified_gmt":"2021-10-10T13:32:07","slug":"what-is-vector-triple-product","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/what-is-vector-triple-product\/","title":{"rendered":"What is Vector Triple Product"},"content":{"rendered":"

Here, you will learn what is vector triple product formula and linear independence and dependence of vectors.<\/p>\n

Let’s begin –<\/p>\n

Vector Triple Product Formula<\/h2>\n

Let \\(\\vec{a}\\), \\(\\vec{b}\\) and \\(\\vec{c}\\) be any three vectors, then the expression<\/p>\n

\n

\\(\\vec{a}\\times (\\vec{b}\\times\\vec{c})\\)<\/p>\n<\/blockquote>\n

is a vector & is called a vector triple product.<\/p>\n

Linear Independence And Dependence of Vectors<\/strong><\/h2>\n

(a)\u00a0 If \\(\\vec{x_1}\\), \\(\\vec{x_2}\\), ……….. \\(\\vec{x_n}\\) are n non zero vectors, & \\(k_1\\), \\(k_1\\), …… \\(k_n\\) are n scalars & if the linear combination \\(k_1\\vec{x_1}\\) + \\(k_2\\vec{x_2}\\) + ….. \\(k_n\\vec{x_n}\\) = \\(\\vec{0}\\) \\(\\implies\\) \\(k_1\\) = 0, \\(k_2\\) = 0 ….. \\(k_n\\) = 0 , then we say that vectors \\(\\vec{x_1}\\), \\(\\vec{x_2}\\), ……….. \\(\\vec{x_n}\\) are linearly independent vectors.<\/p>\n

(b)\u00a0 If \\(\\vec{x_1}\\), \\(\\vec{x_2}\\), ……….. \\(\\vec{x_n}\\) are not linearly independent then they are said to be linearly dependent vectors. i.e. if \\(k_1\\vec{x_1}\\) + \\(k_2\\vec{x_2}\\) + ….. \\(k_n\\vec{x_n}\\) = \\(\\vec{0}\\) & if there exists at least one \\(k_r\\) \\(\\ne\\) 0 then \\(\\vec{x_1}\\), \\(\\vec{x_2}\\), ……….. \\(\\vec{x_n}\\) are said to be linearly dependent.<\/p>\n

Fundamental Theorem in Space<\/strong><\/h4>\n

Let \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) be non-zero, non-coplanar vectors in space. Then any vector \\(\\vec{r}\\), can be uniquely expressed as a linear combination of \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) i.e. There exist some unique x, y, z \\(\\in\\) R such that \\(\\vec{r}\\) = \\(x\\vec{a}\\) + \\(y\\vec{b}\\) + \\(z\\vec{c}\\)<\/p>\n

Shortest Distance Between Two Lines<\/h2>\n

If two lines in space intersect at a point, then obviously the shortest distance between them is zero. Lines which do not intersect & also are not parallel are called skew lines. In other words the lines which are not coplanar are skew lines.<\/p>\n

If two lines are given by\u00a0 \u00a0\\(\\vec{r_1}\\) = \\(\\vec{a_1}\\) + \\(K_1\\vec{b_1}\\) & \\(\\vec{r_2}\\) = \\(\\vec{a_2}\\) + \\(K_2\\vec{b_2}\\)\u00a0 \u00a0then shortest distance between two lines are given by :<\/p>\n

\n

d = |\\((\\vec{a_1} – \\vec{a_1}).(\\vec{b_1} \\times \\vec{b_2})\\over |\\vec{b_1} \\times \\vec{b_2}|\\)|<\/p>\n<\/blockquote>\n

If two lines are given by\u00a0 \\(\\vec{r_1}\\) = \\(\\vec{a_1}\\) + \\(K_1\\vec{b}\\) & \\(\\vec{r_2}\\) = \\(\\vec{a_2}\\) + \\(K_2\\vec{b}\\) i.e. they are parallel, then shortest distance between two lines are given by :<\/p>\n

\n

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

\n
Previous – What is Scalar Triple Product \u2013 Properties and Examples<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here, you will learn what is vector triple product formula and linear independence and dependence of vectors. Let’s begin – Vector Triple Product Formula Let \\(\\vec{a}\\), \\(\\vec{b}\\) and \\(\\vec{c}\\) be any three vectors, then the expression \\(\\vec{a}\\times (\\vec{b}\\times\\vec{c})\\) is a vector & is called a vector triple product. Linear Independence And Dependence of Vectors (a)\u00a0 …<\/p>\n

What is Vector Triple Product<\/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":"\nWhat is Vector Triple Product - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is vector triple product and linear independence and dependence of vectors.\" \/>\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\/what-is-vector-triple-product\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What is Vector Triple Product - 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