If a unit vector $\vec{a}$ makes angles $\pi / 3$ with $\hat{i}, \pi / 4$ with $\hat{j}$ and $ heta

Question

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

Professor PC Thomas Classes · Answer

(b) : We know,\ $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$\ $\Rightarrow\left(\cos \frac{\pi}{3}ight)^{2}+\left(\cos \frac{\pi}{4}ight)^{2}+\cos ^{2} 	heta=1$ $$ \left[\because 	ext { Given } \alpha=\frac{\pi}{3}, \beta=\frac{\pi}{4} 	ext { and } \gamma=	hetaight] $$ $\Rightarrow \frac{1}{4}+\frac{1}{2}+\cos ^{2} 	heta=1$\ $\Rightarrow \cos ^{2} 	heta=1-\frac{3}{4}=\frac{1}{4}$\ $\Rightarrow \cos 	heta= \pm \frac{1}{2} \quad 	herefore \quad 	heta=\frac{\pi}{3}$ or $\frac{2 \pi}{3}$\