Position vectors and division of a segment

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

Example

Point \(P\) divides \(AB\) internally with \(AP:PB=2:3\). The whole segment contains five ratio parts, so \(\overrightarrow{AP}=\dfrac25(\mathbf b-\mathbf a)\).

\(\begin{aligned}\overrightarrow{OP}&=\mathbf a+\dfrac25(\mathbf b-\mathbf a)\\&=\dfrac35\mathbf a+\dfrac25\mathbf b.\end{aligned}\)

For \(A=(1,2)\), \(B=(11,7)\), \(\overrightarrow{AP}=\begin{pmatrix}4\\2\end{pmatrix}\), so \(P=(5,4)\). The remaining displacement is \(\begin{pmatrix}6\\3\end{pmatrix}\), giving the required ratio \(2:3\).

The fraction travelled is \(\dfrac{AP}{AB}=\dfrac25\), not \(\dfrac23\). At a midpoint the fraction is \(\dfrac12\), giving position vector \(\dfrac12(\mathbf a+\mathbf b)\).