Review question

# If $PRBQ$ is a parallelogram, what is the position vector of $Q$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8573

## Solution

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 $B$.