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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}