If a unit vector $\overrightarrow{a}$ makes angles $\frac{\pi }{3}$ with $\hat{i}, \frac{\mathrm{\pi }}{4}$ with $\widehat{j}$ and $\theta \in \left(0,\pi \right)$ with $\widehat{k},$ then a value of $\theta$ is:

The unit vector $\overrightarrow{a}$ makes angle $\frac{\mathrm{\pi }}{3}, \frac{\mathrm{\pi }}{4}$ and $\theta$ with $\hat{i}, \hat{j} \& \hat{k}$ respectively.

Hence, the direction cosines of the vector $\overrightarrow{a}$ are $\cos \alpha =\cos \frac{\mathrm{\pi }}{3}=\frac{1}{2}, \cos \beta =\cos \frac{\mathrm{\pi }}{4}=\frac{1}{\sqrt{2}}$ and $\cos \gamma =\cos \theta$

For a line having direction cosines as $\cos \alpha , \cos \beta , \cos \gamma$

$\Rightarrow {\mathrm{c}\mathrm{o}\mathrm{s}}^2\alpha +{\mathrm{c}\mathrm{o}\mathrm{s}}^2\beta +{\mathrm{c}\mathrm{o}\mathrm{s}}^2\gamma =1$

$\Rightarrow \frac{1}{4}+\frac{1}{2}+{\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta =1$

$\Rightarrow {\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta =1-\frac{3}{4}=\frac{1}{4}$

$\Rightarrow {\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta =\pm \frac{1}{2}$

$\Rightarrow \theta =\frac{\pi }{3}$ or $\frac{2\pi }{3}.$