Find the value of $x$ for which $x(\hat i + \hat j + \hat k)$ is a unit vector.

Let us take $\vec a = x\hat i + x\hat j + x\hat k$. As we know, $|\vec a| = 1$

$\Rightarrow \sqrt {{x^2} + {x^2} + {x^2}} = 1 \Rightarrow \sqrt 3 x = \pm 1 \Rightarrow x = \pm \cfrac{1}{{\sqrt 3 }}$