The points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively, relative to an origin \(O\). The point \(P\) divides the line segment \(OA\) in the ratio \(1:3\), and the point \(R\) divides the line segment \(AB\) in the ratio \(1:2\). Given that \(PRBQ\) is a parallelogram, determine the position vector of \(Q\).
For convenience, we write \(\mathbf{p}\) and \(\mathbf{q}\) for the position vectors of \(P\) and \(Q\).
Since \(PRBQ\) is a parallelogram, \(\overrightarrow{PQ} = \overrightarrow{RB}\).
Using the ratios given in the question we have, \[\begin{align*} \mathbf{p} &= \frac{1}{4} \mathbf{a} \\ \text{and}\quad \mathbf{q} &= \mathbf{p}+\overrightarrow{PQ} = \mathbf{p}+\overrightarrow{RB} = \frac{1}{4} \mathbf{a} +\frac{2}{3}(\mathbf{b}-\mathbf{a}) \\ &= \dfrac{2}{3}\mathbf{b}-\dfrac{5}{12}\mathbf{a}. \end{align*}\]The GeoGebra applet below allows us to explore the behaviour as we move \(A\) and \(B\).
