Question

A vector $\overset{ \rightarrow }{r}$ is equally inclined with the vectors $\overset{ \rightarrow }{a}=cos \theta \hat{i}+sin⁡\theta \hat{j},\overset{ \rightarrow }{b}=-sin⁡\theta \hat{i}+cos⁡\theta \hat{j}$ and $\overset{ \rightarrow }{c}=\hat{k},$ then the angle between $\overset{ \rightarrow }{r}$ and $\overset{ \rightarrow }{a}$ is

• A. $cos^{- 1}\left(\frac{1}{\sqrt{2}}\right)$
• B. $cos^{- 1}\left(\frac{1}{3}\right)$
• C. $cos^{- 1}\left(\frac{1}{\sqrt{3}}\right)$
• D. $cos^{- 1}\frac{1}{2}$

Answer 1

Answer: (C)

$\overset{ \rightarrow }{a},\overset{ \rightarrow }{b},\overset{ \rightarrow }{c}$ are unit vectors and mutually perpendicular vectors and $\overset{ \rightarrow }{r}$ is equally inclined to all $3$ vectors
$\Rightarrow \overset{ \rightarrow }{r}=\lambda \left(\overset{ \rightarrow }{a} + \overset{ \rightarrow }{b} + \overset{ \rightarrow }{c}\right)$
$\Rightarrow \overset{ \rightarrow }{r}=\lambda \left(\left(cos \theta - sin ⁡ \theta \right) \hat{i} + \left(sin ⁡ \theta + cos ⁡ \theta \right) \hat{j} + \hat{k}\right)$
Angle between $\overset{ \rightarrow }{r}$ and $\overset{ \rightarrow }{a}$ is
$cos \alpha =\frac{\overset{ \rightarrow }{r} \cdot \overset{ \rightarrow }{a}}{\left|\overset{ \rightarrow }{r}\right| \left|\overset{ \rightarrow }{a}\right|}=\frac{\lambda \left(\left(cos\right)^{2} ⁡ \theta - sin ⁡ \theta cos ⁡ \theta + \left(sin\right)^{2} ⁡ \theta + sin ⁡ \theta cos ⁡ \theta \right)}{1 \times \sqrt{3} \lambda }$
$cos \alpha =\frac{1}{\sqrt{3}}\Rightarrow \alpha =cos^{- 1}\left(\frac{1}{\sqrt{3}}\right)$