Show that the direction cosines of a vector equally inclined to the axes

Let a vector be equally inclined to axes OX, OY, and OZ at angle α.

Then, the direction cosines of the vector are cos α, cos α, and cos α.

Now,

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

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

$\Rightarrow \cos \alpha=\frac{1}{\sqrt{3}}$

Hence, the direction cosines of the vector which are equally inclined to the axes are $\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$.