Question

A line $AB$ in three dimensional space makes angles ${45}^{\circ }$ and ${120}^{\circ }$ with the positive $x$ and $y-$axis respectively. If $AB$ makes an acute angle $\theta$ with the positive $z-$axis. Then $\theta$ equals:

• A. ${30}^{\circ }$
• B. ${45}^{\circ }$
• C. ${60}^{\circ }$
• D. ${75}^{\circ }$

Answer 1

The correct option is C ${60}^{\circ }$

$\alpha ={45}^{\circ },\beta ={120}^{\circ },\gamma =\theta$
but ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$
$\Rightarrow \frac{1}{2}+\frac{1}{4}+{\cos }^2\theta =1$
$\Rightarrow {\cos }^2\theta =\frac{1}{4}\Rightarrow \cos \theta =\frac{1}{2}\Rightarrow \theta ={60}^{\circ }$