A unit vector $\hat{a}$ makes angles $\frac{\pi}{3}$ with $\hat{i}, \frac{\pi}{4}$ with $\hat{j}$ and $ heta \in( heta, \pi)$ with $\widehat{k}$, then a value of $ heta$ is

Question

A unit vector $\hat{a}$ makes angles $\frac{\pi}{3}$ with $\hat{i}, \frac{\pi}{4}$ with $\hat{j}$ and $	heta \in(	heta, \pi)$ with $\widehat{k}$, then a value of $	heta$ is

Aicharya · Accepted Answer

$\begin{aligned} & \alpha=\frac{\pi}{3}, \beta=\frac{\pi}{4} 	ext { and } \gamma=0 \ & \because \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1 \ & \Rightarrow \cos ^2 \frac{\pi}{3}+\cos ^2 \frac{\pi}{4}+\cos ^2 	heta=1 \ & \Rightarrow \frac{1}{4}+\frac{1}{2}+\cos ^2 	heta=1\end{aligned}$ $\Rightarrow \cos ^2 	heta=\frac{1}{4}$ $\Rightarrow \cos 	heta= \pm \frac{1}{2}$ $\Rightarrow 	heta=\frac{\pi}{3}$ or $\frac{2 \pi}{3}$ as $	heta \in(0, \pi)$

A unit vector $\hat{a}$ makes angles $\frac{\pi}{3}$ with $\hat{i}, \frac{\pi}{4}$ with $\hat{j}$ and…

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