Question

Let $\overline{a},\overline{b}$ be unit vectors, $\overline{a}\perp \overline{b}$ If $\overline{r}$ is a vector such that $|\overline{r}|=2,(\overline{a},\overline{r})=\frac{\pi }{3}$ and $(\overline{a}\times \overline{b})\times (\overline{a}\times \overline{r})=\overline{a}$ Then $(\overline{b},\overline{r})=$

• A. $\frac{\pi }{4}$
  
• B. $\frac{\pi }{6}$
  
• C. $\frac{\pi }{3}$
  
• D. $\frac{\pi }{2}$

Answer 1

The correct option is A $\frac{\pi }{4}$


$(\overline{a}\times \overline{b})\times (\overline{a}\times \overline{r})=\overline{a}\Rightarrow \overline{a}\times \overline{b}.\overline{r}=1|\overline{a}\times \overline{b}||\overline{r}|cos\theta =1\Rightarrow \theta ={60}^{\circ }\therefore (\overrightarrow{r},\overrightarrow{a}\times \overrightarrow{b})={60}^{\circ },(\overrightarrow{r},\overrightarrow{a})={60}^{\circ }$
Let $(\overrightarrow{r},\overrightarrow{b})=\alpha$
$co{s}^2{60}^{\circ }+co{s}^2{60}^{\circ }+co{s}^2\alpha =1\Rightarrow \alpha =\frac{\pi }{4}$