Question

If $a,b$ and $c$ be three unit vectors such that$a\times (b\times c)=1/2b$, $b$and $c$ being non - parallel. If ${\theta }_1$ is the angle between $a$ and $b$ and ${\theta }_2$ is the angle between $a$ and $c$, then

• A. ${\theta }_1=\pi /6,{\theta }_2=\pi /3$
• B. ${\theta }_1=\pi /3,{\theta }_2=\pi /6$
• C. ${\theta }_1=\pi /2,{\theta }_2=\pi /3$
• D. ${\theta }_1=\pi /3,{\theta }_2=\pi /2$

Answer 1

The correct option is C

${\theta }_1=\pi /2,{\theta }_2=\pi /3$

Explanation for the correct option:

Step1. Find the angle between $a$and $c$:

Given $ax(bxc)=\frac{1}{2}b$

$\Rightarrow (a.c)b–(a.b)c=\frac{1}{2}b$

It is possible only if $a\cdot c=1/2$ and $a\cdot b=0$ as $b$ and $c$ are non-parallel.

Since ${\theta }_1$ is the angle between $a$ and $b$ and ${\theta }_2$ is the angle between $a$ and $c$.

The angle between $a$and $c$ is given by

$$\begin{gathered} \Rightarrow \cos {\theta }_2=\frac{(a.c)}{\left|a\right|\left|c\right|} \\ \Rightarrow \cos {\theta }_2=\frac{1}{2} \\ \Rightarrow \cos {\theta }_2=\cos (\pi /3) \\ \Rightarrow {\theta }_2=\pi /3 \end{gathered}$$

Step2. Find the angle between $a$and $b$:

The angle between $a$and $b$ is given by

$$\begin{gathered} \Rightarrow \cos {\theta }_1=\frac{(a.b)}{\left|a\right|\left|c\right|} \\ \Rightarrow \cos {\theta }_1=\frac{{\displaystyle 0}}{\left|a\right|\left|c\right|} \\ \Rightarrow \cos {\theta }_1=0 \\ \Rightarrow \cos {\theta }_1=\cos (\pi /2) \\ \Rightarrow {\theta }_1=\pi /2 \end{gathered}$$

Hence, the correct option is (C).