If a unit vector $\vec a$ makes angles $\cfrac{\pi }{3}$ with $\hat i,\cfrac{\pi }{4}$with $\hat j$ and an acute angle $\theta$ with $\hat k$ , then find $\theta$ and hence, the components of $\vec a$.

Here, ${\cos ^2}\cfrac{\pi }{3} + {\cos ^2}\cfrac{\pi }{4} + {\cos ^2}\theta = 1$
$\Rightarrow {\left( {\cfrac{1}{2}} \right)^2} + {\left( {\cfrac{1}{{\sqrt 2 }}} \right)^2} + {\cos ^2}\theta = 1$

$\Rightarrow \cfrac{1}{4} + \cfrac{1}{2} + {\cos ^2}\theta = 1 \Rightarrow {\cos ^2}\theta = \cfrac{1}{4}$

$\Rightarrow \cos \theta = \cfrac{1}{2}$ (Take$+ ve$ sign) $\Rightarrow \theta = \cfrac{\pi }{3}$

Components of $\vec a$ are $\cos \cfrac{\pi }{3},\cos \cfrac{\pi }{4},\cos \cfrac{\pi }{3}$ i.e.,

$\cfrac{1}{2},\cfrac{1}{{\sqrt 2 }},\cfrac{1}{2}$