Question

If a line makes ${60}^o$ & ${45}^o$ with +ve x-axis and +ve y axis respectively find the angle between line and +ve z-axis.

Answer 1

If $\alpha ,\beta ,\gamma$ are the angles made by a line with the positive direction of $x-$axis,$y-$axis and $z-$axis then

$\cos \alpha ,\cos \beta ,\cos \gamma$ are called the direction cosines of the line

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

Given: $\alpha ={45}^{\circ }$ and $\beta ={60}^{\circ }$

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

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

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

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

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