Example 3. Find the unit vector and direction cosines of the vector $\vec{a}=\{6,-2,-3\}$.
Solution. We have $|\vec{a}|=\sqrt{36+4+9}=7$. According to points $3^{\circ}, 4^{\circ}$, $\vec{e}_{a}=\left\{\frac{6}{7},-\frac{2}{7},-\frac{3}{7}\right\}$ and the direction cosines of vector $\vec{a}: \cos \alpha=\frac{6}{7}$, $\cos \beta=-\frac{2}{7}, \cos \gamma=-\frac{3}{7}$, and $\sqrt{\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma}=\left|\vec{a}_{\vec{a}}\right|=1$.