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  4. If a unit vector veca makes angles π/3 with hati , π /4 with hatj and θ ∈ (0 , π) with hatk, then a value of θ is: -

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Q. 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 : -

JEE MainJEE Main 2019Vector Algebra

Solution:

$\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} $