Question

If a unit vector $\overrightarrow{a}$ makes angles $\frac{\pi }{3}$ with $\hat{i}$, $\frac{\pi }{4}$ with $\hat{j}$ and an acute angle $\theta$ with $\hat{k}$, then find $\theta$ and hence, the components of $\overrightarrow{a}$

Answer 1

Let $\alpha =\frac{\pi }{3},\beta =\frac{\pi }{4},\gamma =\theta$ be the angles made with $\hat{i}\hat{j}\hat{k}$ axes.

$\therefore \cos {}^2\alpha +\cos {}^2\beta +\cos {}^2\gamma =1\Rightarrow \cos {}^2\gamma =1-\cos {}^2\frac{\pi }{3}-\cos {}^2\frac{\pi }{4}\Rightarrow \cos {}^2\gamma =\frac{1}{4}\Rightarrow \gamma =\frac{\pi }{3}$

$\therefore$ the acute angle $\theta$ that it makes with $\hat{k}$ is $\frac{\pi }{3}$.

Hence,components of $\overrightarrow{a}$ along $\hat{i},\hat{j},\hat{k}$ are $\hat{i},\hat{j},\hat{k}$

$\Rightarrow \frac{1}{2},\frac{1}{\sqrt{2}},\frac{1}{2}$