Question

If $b$ and $c$ are any two non-collinear unit vecators and $a$ is nay vector, then $(a.b)b+(a.c)c+\frac{a.(b+c)}{{|b+c|}^2}(b\times c)=$

• A. $a$
• B. $b$
• C. $c$
• D. none of these

Answer 1

Answer: (A)

$a$

Let $I$ be $a$ unit vector in the direction of $b,J$ in the direction of $c.$

Note that $b=1$ and $c=J.$

We have $b\times c=\left|b\right|\left|c\right|\sin \alpha K,$

Where $K$ is a unit vector perpendicular to $b$ and $c$.

$\Rightarrow |b\times c|=\sin \alpha \Rightarrow k=\frac{b\times c}{|b\times c|}$

Any vector $a$ can be written as $a$ linear combination of $I,J$ and $K.$

Let $a=a_{1}I+a_{2}J+a_{3}K$

Now, $a.b=a.I=a_1,a.c=a.J=a_2$

and, $a.\frac{b\times c}{|b\times c|}=a.K=a_3$

Thus, $(a.b)b+(a.c)c+\frac{a.(b\times c)}{{|b\times c|}^2}(b\times c)$

$=a_{1}b+a_{2}c+a_3\frac{b\times c}{|b\times c|}$

$=a_{1}I+a_{2}J+a_{3}K=a.$