Question

If a line makes angles ${90}^{\circ },{135}^{\circ },{45}^{\circ }$ with the $x,y\text{and}z-\text{axis}$ respectively, find its direction cosines.

Answer 1

$\left[\begin{array}{c}\text{Direction cosines of the line making angle}\\ \alpha ,\beta \text{and}\gamma \text{with}x,y\text{and}z-\text{axis are}l,m,n.\\ l=\cos \alpha ,m=\cos \beta ,n=\cos \gamma \end{array}\right]$

Here, $\alpha ={90}^{\circ },\beta ={135}^{\circ },\gamma ={45}^{\circ }$

So, direction cosines are:

$l=\cos {90}^{\circ }=0,$

$m=\cos {135}^{\circ }=-\frac{1}{\sqrt{2}},$

$n=\cos {45}^{\circ }=\frac{1}{\sqrt{2}}$

$\therefore$ Required direction cosines are: $0,-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}$