Question

If $b$ and $c$ are any two non-collinear unit vectors and $a$ is any vector, then
$(a.b)b+(a.c)+\frac{a.b\times c}{|b\times c|}.(b\times c)$ is equal to

• A. $0$
• B. $a$
• C. $b$
• D. $c$

Answer 1

The correct option is B $a$

Let $b=\hat{i},c=\hat{j}$
$\therefore |b\times c|=\left|\hat{k}\right|=1$
again, let $a=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$
Now, $a.b=a.\hat{i}=a_1,a.\overline{c}=a.\hat{j}=a_2$
and $a.\frac{b.c}{|b\times c|}=a.\hat{k}=a_3$
$\therefore (a.b)b+(a.c)c+a.\frac{b.c}{|b\times c|}(b\times c)$
$=a_{1}b+a_{2}c+a_3(b\times c)=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}=a$