{"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
\nV = [\\(\\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>\n10).<\/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\nExample : <\/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\nHope 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