Normalizing a Vector

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

1. \(\mathbf{v} = \langle 3,-4\rangle\)
2. \(\mathbf{v} = \langle -1,2,-2\rangle\)

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 \[ \begin{eqnarray*} \mathbf{u} = \dfrac{\mathbf{v}}{\Vert \mathbf{v\Vert }} = \dfrac{\mathbf{v}}{3} = \left\langle -\dfrac{1}{3}, \dfrac{2}{3}, -\dfrac{2}{3}\right\rangle \end{eqnarray*} \]