Solution (a) The magnitude of $\mathbf{v}$ is $$\begin{equation*} \Vert \mathbf{v}\Vert =\sqrt{(-3)^{2}+2^{2}+(-6)^{2}}=\sqrt{49}=7 \end{equation*}$$

The direction cosines of the vector $\mathbf{v}$ are $$\cos \alpha =\dfrac{v_{1}}{\left\Vert \mathbf{v}\right\Vert }=\dfrac{-3}{7} \qquad \cos \beta =\dfrac{v_{2}}{\left\Vert \mathbf{v}\right\Vert }= \dfrac{2}{7} \qquad \cos \gamma =\dfrac{v_{3}}{\left\Vert \mathbf{v}\right\Vert }=\dfrac{-6}{7}$$

(b) Now $\mathbf{v=}\left\Vert \mathbf{v}\right\Vert \left[ \cos \alpha {\bf i} + \cos \beta \mathbf{j}+\cos \gamma \mathbf{k}\right] =7\!\left( -\dfrac{3}{7}\mathbf{i}+\dfrac{2}{7}\mathbf{j}-\dfrac{6}{7}\mathbf{k }\right)\! .$