Precalculus (6th Edition) Blitzer

The unit vector that has the same direction as vector $\mathbf{v}$ is given by $\frac{v}{\left\| v \right\|}=\frac{a\mathbf{i}+b\mathbf{j}}{\sqrt{{{a}^{2}}+{{b}^{2}}}}$.
Let $\mathbf{v}$ be any non-zero vector given as $\mathbf{v}=a\mathbf{i}+b\mathbf{j}$. In order to find the unit vector that has the same direction, first, calculate $\left\| \mathbf{v} \right\|$. $\left\| \mathbf{v} \right\|=\sqrt{{{a}^{2}}+{{b}^{2}}}$ Then the unit vector that has the same direction as vector $\mathbf{v}$ is given by $\frac{v}{\left\| v \right\|}=\frac{a\mathbf{i}+b\mathbf{j}}{\sqrt{{{a}^{2}}+{{b}^{2}}}}$ …… (1) Example: Information: Let vector $\mathbf{v}=2\mathbf{i}+3\mathbf{j}$. Calculate $\left\| \mathbf{v} \right\|$ as follows: \begin{align} & \left\| \mathbf{v} \right\|=\sqrt{{{a}^{2}}+{{b}^{2}}} \\ & =\sqrt{{{2}^{2}}+{{3}^{2}}} \\ & =\sqrt{13} \end{align} Putting the above value in equation (1) gives $\frac{v}{\left\| v \right\|}=\frac{2\mathbf{i}+3\mathbf{j}}{\sqrt{13}}$