The direction cosines of a vector $\vec{r}$ which is equally inclined with OX, OY and OZ, are

$±\frac{1}{\sqrt{3}}, ±\frac{1}{\sqrt{3}}, ±\frac{1}{\sqrt{3}}$

Let l, m, n be the direction cosines of $\vec{r}$. Since $\vec{r}$ is equally inclined with OX, OY and OZ.

$∴ l = m = n $       $[∵ \alpha = \beta = \gamma ⇒ cos \alpha = cos \beta = cos \gamma ]$

Now, $l^2 + m^2 + n^2 = 1 ⇒ 2l^2 = 1 ⇒ l = ±\frac{1}{\sqrt{3}}$

Hence, direction cosines of $vec{r}$ are  $±\frac{1}{\sqrt{3}}, ±\frac{1}{\sqrt{3}}, ±\frac{1}{\sqrt{3}}$