Question

If a unit vector $\vec{a}$ makes angles $\pi/3$ with $\hat{i} , \pi /4$ with $\hat{j}$ and $\theta \in (0 , \pi)$ with $\hat{k}$, then a value of $\theta$ is : -

• A. $\frac{5 \pi}{12}$
• B. $\frac{5 \pi}{6}$
• C. $\frac{2 \pi}{3}$
• D. $\frac{\pi}{4}$

Answer 1

Answer: (C)

$\frac{2 \pi}{3}$

Answer 2

$\cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma = 1$
$\frac{1}{4} + \frac{1}{2} + \cos^{2} \gamma = 1$
$\cos^{2} \gamma = 1 - \frac{3}{4} = \frac{1}{4}$
$\cos^{2} \gamma = \pm \frac{1}{2} \Rightarrow \gamma = \frac{\pi}{3}$ or $\frac{2\pi}{3}$