Question

If $l,m,n$ are the direction cosines of a line, then prove that $l^2+m^2+n^2=1$. Hence find the direction angle of the line with the $X$ axis which makes direction angles of ${135}^{\circ }$ and ${45}^{\circ }$ with $Y$ and $Z$ axes respectively

Answer 1

Let the line makes angle $\alpha$ with x-axis, $\beta$ with y-axis and $\gamma$ with z-axis, then

$l^2+m^2+n^2=1$ or ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$

Here given $\alpha ={135}^o$ and $\beta ={45}^o$ and $\gamma =?$

So ${\cos }^2{135}^0+{\cos }^2{45}^o+{\cos }^2\gamma =1$

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

$\Rightarrow \cos \gamma =0\Rightarrow \gamma ={90}^o$

Hence given line makes angle ${90}^o$ with the z-axis