# Position Vectors & Distances

A position vector is any vector that is placed extending from the origin. Position vectors are often denoted by $\overrightarrow{OA}$, for example, to identify the vector that points from the origin to a point A.

Let $\overrightarrow{OA}$ and $\overrightarrow{OB}$ be the position vectors that point from the origin to the points A and B respectively. We can then find the vector that points from A to B:

$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$

The length of a vector can be found using pythagoras.

Example – If

$\overrightarrow{OA}=\left(\begin{array}{c}2\\-4\end{array}\right)$, $\overrightarrow{OB}=\left(\begin{array}{c}3\\1\end{array}\right)$

then

$\overrightarrow{AB}=\left(\begin{array}{c}3\\1\end{array}\right)-\left(\begin{array}{c}2\\-4\end{array}\right)=\left(\begin{array}{c}1\\5\end{array}\right)$.

The length of this vector can then be found using pythagoras:

$\vert\overrightarrow{AB}\vert=\sqrt{1^2+5^2}=\sqrt{26}$