If $PRBQ$ is a parallelogram, what is the position vector of $Q$?

If $PRBQ$ is a parallelogram, what is the position vector of $Q$?

Question

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 $Q$.

Solution

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 $B$.

[interactive embed (geogebra-iframe): //www.geogebratube.org/material/iframe/id/tpe4c9Un/width/600/height/317/border/888888/rc/false/ai/false/sdz/false/smb/false/stb/false/stbh/false/ld/false/sri/true/asb/false/at/html5] — not part of the text import; see the source page.

Download GeoGebra file