Position vectors and directed routes

A position vector starts at a fixed origin. If \(\overrightarrow{OA}=\mathbf{a}\) and \(\overrightarrow{OB}=\mathbf{b}\), then the route \(A\to O\to B\) gives \(\overrightarrow{AB}=-\mathbf{a}+\mathbf{b}=\mathbf{b}-\mathbf{a}\).

Vector routes join head to tail: \(\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}\). The order of the letters determines the direction, so \(\overrightarrow{BA}=-\overrightarrow{AB}\).

In a parallelogram, opposite directed sides are equal when their arrows point the same way. Draw the route before simplifying; a diagram’s apparent length or position is not an extra given.