Answer
Position vector for two points is given as $\mathbf{v}=\left( {{x}_{2}}-{{x}_{1}} \right)\mathbf{i}+\left( {{y}_{2}}-{{y}_{1}} \right)\mathbf{j}$.
Work Step by Step
A position vector is a vector whose initial point is at the origin.
Any vector in rectangular coordinates can be represented as a position vector.
Consider the vector $\mathbf{v}$ with initial point ${{P}_{1}}=\left( {{x}_{1}},{{y}_{1}} \right)$ and terminal point ${{P}_{2}}=\left( {{x}_{2}},{{y}_{2}} \right)$.
Then the vector $\mathbf{v}$ is equal to the position vector.
$\mathbf{v}=\left( {{x}_{2}}-{{x}_{1}} \right)\mathbf{i}+\left( {{y}_{2}}-{{y}_{1}} \right)\mathbf{j}$
Example: Information:
Consider the vector $\mathbf{v}$ with initial point ${{P}_{1}}=\left( 4,3 \right)$ and terminal point ${{P}_{2}}=\left( 8,5 \right)$.
Then, the position vector $\mathbf{v}$ is given as
$\begin{align}
& \mathbf{v}=\left( {{x}_{2}}-{{x}_{1}} \right)\mathbf{i}+\left( {{y}_{2}}-{{y}_{1}} \right)\mathbf{j} \\
& =\left( 8-4 \right)\mathbf{i}+\left( 5-3 \right)\mathbf{j} \\
& =4\mathbf{i}+2\mathbf{j}
\end{align}$
The position vector $\mathbf{v}$ is $4\mathbf{i}+2\mathbf{j}$.
