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A unit vector $\hat{a}$ makes angles $\frac{\pi}{3}$ with $\hat{i}, \frac{\pi}{4}$ with $\hat{j}$ and $\theta \in(\theta, \pi)$ with $\widehat{k}$, then a value of $\theta$ is
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The correct answer is:
$\frac{2 \pi}{3}$
$\begin{aligned} & \alpha=\frac{\pi}{3}, \beta=\frac{\pi}{4} \text { and } \gamma=0 \\ & \because \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1 \\ & \Rightarrow \cos ^2 \frac{\pi}{3}+\cos ^2 \frac{\pi}{4}+\cos ^2 \theta=1 \\ & \Rightarrow \frac{1}{4}+\frac{1}{2}+\cos ^2 \theta=1\end{aligned}$
$\Rightarrow \cos ^2 \theta=\frac{1}{4}$
$\Rightarrow \cos \theta= \pm \frac{1}{2}$
$\Rightarrow \theta=\frac{\pi}{3}$ or $\frac{2 \pi}{3}$ as $\theta \in(0, \pi)$
$\Rightarrow \cos ^2 \theta=\frac{1}{4}$
$\Rightarrow \cos \theta= \pm \frac{1}{2}$
$\Rightarrow \theta=\frac{\pi}{3}$ or $\frac{2 \pi}{3}$ as $\theta \in(0, \pi)$
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