Normalize each vector. That is, find a unit vector $\mathbf{u}$ that has the same direction as:

Solution (a) Since $\mathbf{v} = \langle 3,-4\rangle$, then $\Vert \mathbf{v}\Vert = \sqrt{9 + 16} = 5$. The unit vector $\mathbf{u}$ in the same direction as $\mathbf{v}$ is $$\begin{eqnarray*} \mathbf{u} = \dfrac{\mathbf{v}}{\Vert \mathbf{v\Vert }} = \dfrac{\mathbf{v}}{5} = \left\langle \dfrac{3}{5}, -\dfrac{4}{5}\right\rangle \end{eqnarray*}$$

(b) Since $\mathbf{v} = \langle -1,2,-2\rangle$, then $\Vert \mathbf{v}\Vert = \sqrt{1 + 4 + 4} = 3$. The unit vector $\mathbf{u}$ in the same direction as $\mathbf{v}$ is $$\begin{eqnarray*} \mathbf{u} = \dfrac{\mathbf{v}}{\Vert \mathbf{v\Vert }} = \dfrac{\mathbf{v}}{3} = \left\langle -\dfrac{1}{3}, \dfrac{2}{3}, -\dfrac{2}{3}\right\rangle \end{eqnarray*}$$