Answer
$\mathbf{v}=3\mathbf{i}-2\mathbf{j}$
Work Step by Step
The vector $\mathbf{v}$ with initial point ${{P}_{1}}=\left( {{x}_{1}},{{y}_{1}} \right)$ and the terminal point ${{P}_{2}}=\left( {{x}_{2}},{{y}_{2}} \right)$ can be written as below:
$\mathbf{v}=\left( {{x}_{2}}-{{x}_{1}} \right)\mathbf{i}+\left( {{y}_{2}}-{{y}_{1}} \right)\mathbf{j}$
Here, $\left( {{x}_{1}},{{y}_{1}} \right)=\left( 2,-1 \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)=\left( 5,-3 \right)$.
So, the vector $\mathbf{v}$ is given by,
$\begin{align}
& \mathbf{v}=\left( {{x}_{2}}-{{x}_{1}} \right)\mathbf{i}+\left( {{y}_{2}}-{{y}_{1}} \right)\mathbf{j} \\
& =\left( 5-2 \right)\mathbf{i}+\left[ -3-\left( -1 \right) \right]\mathbf{j} \\
& =3\mathbf{i}-2\mathbf{j}
\end{align}$
