Question

If a line makes ${90}^{\circ }$ and ${60}^{\circ }$ with x & y axis, find angle it make with z-axis.

Answer 1

Let say angle with x - axis is $\alpha ={90}^o$

angle with y - axis is $\beta ={60}^o$

angle with z - axis is $\gamma =\gamma$

Now we know that sum of direction cosines's square is equal to '1'. It means -

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

${\cos }^2{90}^o+{\cos }^2{60}^o+{\cos }^2\gamma =1$

$0+\frac{1}{4}+{\cos }^2\gamma =1$

${\cos }^2\gamma =1-\frac{1}{4}=\frac{3}{4}$

$\cos \gamma =\pm \frac{\sqrt{3}}{2}$

So, $\gamma ={30}^o$

or, $\gamma ={150}^o$

Angle with z -axis is ${30}^o$ or ${150}^o$