Question

For three vectors $a,b,c[a\times bb\times cc\times a]$ is equal to

• A. $\left[abc\right]$
• B. $[bac{]}^2$
• C. $0$
• D. $2\left[abc\right]$

Answer 1

The correct option is D $2\left[abc\right]$

Given problem is in the form scalar triple product

$\begin{array}{c}\left[a\times bb\times cc\times a\right]=\left(a\times b\right)\times \left(b\times c\right)\cdot \left(c\times a\right)\\ \\ =\left(a\times b+b\times b+a\times c+b\times c\right)\cdot \left(c\times a\right)\\ \\ =\left(a\times b+a\times c+b\times c\right)\cdot \left(c\times a\right)\left(b\times b=0\right)\\ \\ =a\times b\cdot c.b\times c\cdot c+a\times c\cdot c.a\times b\cdot a+b\times c\cdot a.a\times c\cdot a\end{array}$

When two vector are involved in the triple product, their, product is equal to $0$.

Therefore,

$\left[abc\right]+\left[abc\right]=2\left[abc\right]$

Hence, option $D$ is correct answer.