Answer
$\alpha \approx57.7^\circ$, $\beta \approx143.3^\circ$, $\gamma \approx74.5^\circ$
Work Step by Step
1. Given $\vec u=<2,-3,1>$, we have
$||\vec u||=\sqrt {4+9+1}=\sqrt {14}$
2. The direction angles:
$\alpha=cos^{-1}\frac{2}{\sqrt {14}}\approx57.7^\circ$
$\beta=cos^{-1}\frac{-3}{\sqrt {14}}\approx143.3^\circ$
$\gamma=cos^{-1}\frac{1}{\sqrt {14}}\approx74.5^\circ$
