Question

If a line makes ${45}^o$, ${60}^o$ with positive direction of axes $x$ and $y$ then the angles it makes with the $z$-axis is

• A. ${30}^o$
• B. ${90}^o$
• C. ${45}^o$
• D. ${60}^o$

Answer 1

The correct option is D ${60}^o$

Fact: If $\alpha ,\beta ,\gamma$ are the angle made by any line with $x,y$ and $z$-axis, then

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

Here given $\alpha ={45}^o,\beta ={60}^o$

$\Rightarrow {\cos }^2{45}^o+{\cos }^2{60}^o+{\cos }^2\gamma =1$

$\Rightarrow \frac{1}{2}+\frac{1}{4}+{\cos }^2\gamma =1$

$\Rightarrow {\cos }^2\gamma =\frac{1}{4}$

$\Rightarrow \cos \gamma =\frac{1}{2}$

$\therefore \gamma ={60}^o$