{"id":3181,"date":"2021-07-20T08:59:34","date_gmt":"2021-07-20T08:59:34","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3181"},"modified":"2021-10-16T18:44:46","modified_gmt":"2021-10-16T13:14:46","slug":"what-is-scalar-triple-product","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/what-is-scalar-triple-product\/","title":{"rendered":"What is Scalar Triple Product – Properties and Examples"},"content":{"rendered":"

Let \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) be three vectors. Then the scalar \\((\\vec{a}\\times \\vec{b}).\\vec{c}\\) is called the scalar triple product of \\(\\vec{a}\\), \\(\\vec{b}\\) and \\(\\vec{c}\\) and is denoted by [\\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)].<\/p>\n

Thus, we have <\/p>\n

\n

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

For three vectors \\(\\vec{a}\\), \\(\\vec{b}\\) & \\(\\vec{c}\\), it is also defined as : (\\(\\vec{a}\\times\\vec{b}\\)).\\(\\vec{c}\\) = \\(|\\vec{a}||\\vec{b}||\\vec{c}|sin\\theta cos\\phi\\) where \\(\\theta\\) is the angle between \\(\\vec{a}\\) & \\(\\vec{b}\\) and \\(\\phi\\) is the angle between \\(\\vec{a}\\) \\(\\times\\) \\(\\vec{b}\\) & \\(\\vec{c}\\).<\/p>\n

Note<\/strong> – It geometrically represents the volume of the parallelopiped whose three coterminous edges are represented by \\(\\vec{a}\\), \\(\\vec{b}\\) & \\(\\vec{c}\\) <\/p>\n

\n

V = [\\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)]<\/p>\n<\/blockquote>\n

Properties of Scalar Triple Product :<\/h2>\n

1).<\/strong> If \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are cyclically permuted the value of scalar triple product remains same.<\/p>\n

i.e.  \\((\\vec{a}\\times \\vec{b}).\\vec{c}\\) = \\((\\vec{b}\\times \\vec{c}).\\vec{a}\\) =\\((\\vec{c}\\times \\vec{a}).\\vec{b}\\)<\/p>\n

or  [\\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)] = [\\(\\vec{b}\\) \\(\\vec{c}\\) \\(\\vec{a}\\)] = [\\(\\vec{c}\\) \\(\\vec{a}\\) \\(\\vec{b}\\)] <\/p>\n

2). <\/strong>The change of cyclic order of vectors in scalar triple product changes the sign of scalar triple product but not the magnitude.<\/p>\n

i.e.  [\\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)] = – [\\(\\vec{b}\\) \\(\\vec{a}\\) \\(\\vec{c}\\)] = – [\\(\\vec{c}\\) \\(\\vec{b}\\) \\(\\vec{a}\\)] = – [\\(\\vec{a}\\) \\(\\vec{c}\\) \\(\\vec{b}\\)]<\/p>\n

3).<\/strong> In scalar triple product the position of dot and cross can be interchanged provided that the cyclic order of the vectors remain same.<\/p>\n

i.e. \\((\\vec{a}\\times \\vec{b}).\\vec{c}\\) = \\(\\vec{a}.(\\vec{b}\\times \\vec{c}\\) <\/p>\n

4).<\/strong> The scalar triple product of three vectors is zero if any two of them are equal or if any two of them are parallel or collinear.<\/p>\n

5). <\/strong>For any three vectors \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) and scalar \\(\\lambda\\), we have<\/p>\n

[\\(\\lambda\\) \\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)] = \\(\\lambda\\) [\\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)]<\/p>\n

6).<\/strong> For any three vectors \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) and three scalars l, m, n<\/p>\n

[\\(l\\vec{a}\\) \\(m\\vec{b}\\) \\(n\\vec{c}\\)] = lmn [\\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)]<\/p>\n

7).<\/strong> The necessary and sufficient condition for three non-zero, non-collinear vectors \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) to be coplanar is that [\\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)] = 0<\/p>\n

i.e. \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are coplanar \\(\\iff\\) [\\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)] = 0<\/p>\n

8).<\/strong> If \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\), \\(\\vec{d}\\) are four vectors, then<\/p>\n

[\\(\\vec{a}\\) + \\(\\vec{b}\\)  \\(\\vec{c}\\)  \\(\\vec{d}\\)] = [\\(\\vec{a}\\) \\(\\vec{c}\\) \\(\\vec{d}\\)] + [\\(\\vec{b}\\) \\(\\vec{c}\\) \\(\\vec{d}\\)]<\/p>\n

9).<\/strong> Let \\(\\vec{a}\\) = \\(a_1\\hat{i} + a_2\\hat{j} + a_3\\hat{k}\\), \\(\\vec{b}\\) = \\(b_1\\hat{i} + b_2\\hat{j} + b_3\\hat{k}\\) and \\(\\vec{c}\\) = \\(c_1\\hat{i} + c_2\\hat{j} + c_3\\hat{k}\\) be three vectors. Then<\/p>\n

[\\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)] = \\(\\begin{vmatrix}
a_1 & a_2 & a_3 \\\\
b_1 & b_2 & b_3 \\\\
c_1 & c_2 & c_3 \\\\
\\end{vmatrix}\\)<\/p>\n

10).<\/strong> (Distributivity of vector product over vector addition) For any three vectors \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\),<\/p>\n

w have \\(\\vec{a}\\times (\\vec{b} + \\vec{c})\\) = \\(\\vec{a}\\times \\vec{b}\\) + \\(\\vec{a}\\times \\vec{c}\\)<\/p>\n

11). <\/strong>If \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are three non-coplanar vectors and \\(\\vec{u}\\), \\(\\vec{v}\\), \\(\\vec{w}\\) are three vectors such that <\/p>\n

\\(\\vec{u}\\) = \\(x_1\\hat{i} + y_1\\hat{j} + z_1\\hat{k}\\)<\/p>\n

\\(\\vec{v}\\) = \\(x_2\\hat{i} + y_2\\hat{j} + z_2\\hat{k}\\)<\/p>\n

\\(\\vec{w}\\) = \\(x_3\\hat{i} + y_3\\hat{j} + z_3\\hat{k}\\)<\/p>\n

Then, [\\(\\vec{u}\\) \\(\\vec{v}\\) \\(\\vec{w}\\)] = \\(\\begin{vmatrix}
x_1 & y_1 & z_1 \\\\
x_2 & y_2 & z_2 \\\\
x_3 & y_3 & z_3 \\\\
\\end{vmatrix}\\) [\\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)] <\/p>\n\n\n

Example : <\/span>Show that \\(\\vec{a}\\) = \\(-2\\hat{i} – 2\\hat{j} + 4\\hat{k}\\), \\(\\vec{b}\\) = \\(-2\\hat{i} + 4\\hat{j} – 2\\hat{k}\\) and \\(\\vec{c}\\) = \\(4\\hat{i} – 2\\hat{j} -2 \\hat{k}\\) are coplanar.<\/p>\n

Solution : <\/span>We know that three vectors \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) are coplanar if [\\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)] = 0.

\nHere, [\\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)] = \\(\\begin{vmatrix}\n-2 & -2 & 4 \\\\\n-2 & 4 & -2 \\\\\n4 & -2 & -2 \\\\\n\\end{vmatrix}\\)

\n-2(-8 – 4) + 2(4 + 8) + 4(4 – 16) = 24 + 24 – 48 = 0

\nHence, the given vectors are coplanar.

\n <\/p>\n\n\n

Hope you learnt what is scalar triple product and its properties, learn more concepts of vectors and practice more questions to get ahead in the competition. Good luck!<\/p>\n\n\n

\n
Next – What is Vector Triple Product \u2013 Coplanarity of Four Points<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Cross Product of Vectors Formula <\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Let \\(\\vec{a}\\), \\(\\vec{b}\\), \\(\\vec{c}\\) be three vectors. Then the scalar \\((\\vec{a}\\times \\vec{b}).\\vec{c}\\) is called the scalar triple product of \\(\\vec{a}\\), \\(\\vec{b}\\) and \\(\\vec{c}\\) and is denoted by [\\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)]. Thus, we have  [\\(\\vec{a}\\) \\(\\vec{b}\\) \\(\\vec{c}\\)] = \\((\\vec{a}\\times \\vec{b}).\\vec{c}\\) For three vectors \\(\\vec{a}\\), \\(\\vec{b}\\) & \\(\\vec{c}\\), it is also defined as : (\\(\\vec{a}\\times\\vec{b}\\)).\\(\\vec{c}\\) = \\(|\\vec{a}||\\vec{b}||\\vec{c}|sin\\theta …<\/p>\n

What is Scalar Triple Product – Properties and Examples<\/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 Scalar Triple Product - Properties and Examples<\/title>\n<meta name=\"description\" content=\"In this post, you will learn what is scalar triple product and properties of scalar triple product 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\/what-is-scalar-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 Scalar Triple Product - 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