## Precalculus (6th Edition) Blitzer

Position vector for two points is given as $\mathbf{v}=\left( {{x}_{2}}-{{x}_{1}} \right)\mathbf{i}+\left( {{y}_{2}}-{{y}_{1}} \right)\mathbf{j}$.
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 $4\mathbf{i}+2\mathbf{j}$.