Show that the direction cosines of a vector equally inclined to the axes $OX,OY$ and $OZ$ are$\cfrac{1}{{\sqrt 3 }},\cfrac{1}{{\sqrt 3 }},\cfrac{1}{{\sqrt 3 }}$.

Let a vector makes $\alpha ,\beta ,\gamma$ angles with axes $OX,OY\,\,{\rm{and}}\,\,OZ$ respectively.

As it is equally inclined to axes,
So, $\alpha = \beta = \gamma$

Now we have
${\cos ^2}a + {\cos ^2}\beta + {\cos ^2}\gamma = 1$

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

$\therefore$

Direction cosines of the vector are $< \cfrac{1}{{\sqrt 3 }}\cfrac{1}{{\sqrt 3 }}\cfrac{1}{{\sqrt 3 }} > + ve\,\,{\rm{sign}}$.