Question

If $a,b,c$ are three non coplanar, non zero vectors, then $(a.a)b\times c+(a.b)c\times a+(a.c)a\times b$ is equal to

• A. $\left[bca\right]a$
• B. $\left[cab\right]b$
• C. $\left[abc\right]c$
• D. none of these

Answer 1

The correct option is A $\left[bca\right]a$

Let $a$ $=l(b\times c)+m(c\times a)+n(a\times b)$ ...(1)

We have to find the values of $l,m$ and $n.$

Multiply both sides of (1) scalarly by $a.$

$a.a=la.(b\times c)+ma.(c\times a)+na.(a\times b)$

$\Rightarrow a.a=l\left[abc\right]+m\left[aca\right]+n\left[aab\right]=l\left[abc\right].$

$\therefore$ Scalar triple product is zero when two vectors are equal

$\therefore l=\frac{a.a}{\left[abc\right]}$

Similarly, multiplying both sides of (1) salarly by $b$ and $c$, we get $m=\frac{a.b}{\left(abc\right)},n=\frac{a.c}{\left[abc\right]}$

$\therefore a=\frac{a.a}{\left[abc\right]}(b\times c)+\frac{a.b}{\left[abc\right]}(c\times a)+\frac{a.c}{\left[abc\right]}(a\times b)$

Similarly we can write the values $b$ and $c$ as $b=\frac{b.a}{\left[abc\right]}(b\times c)+\frac{b.b}{\left[abc\right]}(c\times a)+\frac{b.c}{\left[abc\right]}(a\times b)$

and $c=\frac{c.a}{\left[abc\right]}(b\times c)+\frac{c.b}{\left[abc\right]}(c\times a)+\frac{c.c}{\left[abc\right]}(a\times b).$

$\therefore a=\frac{(a.a)b\times c}{\left[bca\right]}+\frac{(a.b)c\times a}{\left[cab\right]}+\frac{(a.c)a\times b}{\left[abc\right]}\therefore (a.a)b\times c+(a.b)c\times a+(a.c)a\times b=\left[bca\right]a$