Q. If the angle between $\widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{k}}$ and $\widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}} + a\widehat{\mathbf{k}}$ is $\frac{\pi}{3}$, then the value of $a$ is
A) 0 or 2
B) $- 4$ or 0
C) 0 or $- 2$
D) 2 or $- 2$
Explanation
(b)
$\cos{\frac{\pi}{3} = \frac{\left( \widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{k}} \right)\mathbf{\bullet (}\widehat{\mathbf{i}} + \widehat{\mathbf{j}} + a\widehat{\mathbf{k}})\ \ }{\sqrt{2}\sqrt{1 + 1 + a^{2}}}}$
$\Rightarrow \frac{1}{2} = \frac{1 + a}{\sqrt{2}\sqrt{2 + a^{2}}}$
$\Rightarrow \frac{1}{4} = \frac{(1 + a)^{2}}{2(2 + a^{2})}$
$\Rightarrow 2 + a^{2} = 2(1 + a^{2} + 2a)$
$\Rightarrow a^{2} + 4a = 0$
$\Rightarrow a = 0,\ - 4$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.