Find the direction cosines of a line which makes equal angles with the coordinate axes.

Let the line makes an angle $\alpha$ with each of the three coordinate axes, then its direction cosines are

$< \cos \alpha ,\cos \alpha ,\cos \alpha >$

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

$\Rightarrow$ $3{\cos ^2}\alpha = 1 \Rightarrow {\cos ^2}\alpha = \cfrac{1}{3} \Rightarrow \cos \alpha = \pm \cfrac{1}{{\sqrt 3 }}$

$\therefore$ Direction cosines of the line are either $< \cfrac{1}{{\sqrt 3 }},\cfrac{1}{{\sqrt 3 }},\cfrac{1}{{\sqrt 3 }} >$

or $< - \cfrac{1}{{\sqrt 3 }}, - \cfrac{1}{{\sqrt 3 }}, - \cfrac{1}{{\sqrt 3 }} >$