Answer
The unit vector that has the same direction as vector $\mathbf{v}$ is given by $\frac{v}{\left\| v \right\|}=\frac{a\mathbf{i}+b\mathbf{j}}{\sqrt{{{a}^{2}}+{{b}^{2}}}}$.
Work Step by Step
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
$\frac{v}{\left\| v \right\|}=\frac{2\mathbf{i}+3\mathbf{j}}{\sqrt{13}}$
