Question

A ray makes angles $\frac{\pi }{3},\frac{\pi }{3}$ with $\overrightarrow{OX}$ and $\overrightarrow{OY}$ respectively. Find the angle made by it with $\overrightarrow{OZ}$.

Answer 1

${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$

${\cos }^2\alpha =$ angle with $\overrightarrow{ox}$

${\cos }^2\beta =$ angle with $\overrightarrow{oy}$

${\cos }^2\gamma =1$ angle with $\overrightarrow{oz}$

${\left(\frac{1}{\alpha }\right)}^2+{\left(\frac{1}{\alpha }\right)}^2+{\cos }^{2}y=1$

${\cos }^{2}y=\frac{2}{4}$

$\cos y=\pm \frac{1}{\sqrt{2}}$

So, angle made with $\overrightarrow{oz}={45}^o$.