Question

A line makes ${45}^{\circ }$ and ${90}^{\circ }$ angles with $X$ and $Y$ -axis respectively. Find the angle it makes with $Z$ -axis.

Answer 1

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

$\frac{1}{2}+0+{\cos }^2\gamma =1$

$\Rightarrow {\cos }^2\gamma =1/2$

$\Rightarrow cos\gamma =\pm \frac{1}{\sqrt{2}}\Rightarrow \gamma ={45}^0$ or ${135}^0$.