$\mathbf{f} = \begin{pmatrix}4\\-3\end{pmatrix}$ and $\mathbf{g} = \begin{pmatrix}1\\-5\end{pmatrix}$.
Points $O, A$ and $B$ are given with $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. Point $P$ lies on $OA$ so that $OP = \frac{1}{3}OA$. The points $O, Q$ and $R$ are collinear, and $Q$ is the midpoint of $PB$.
(a(i))[1]
Find the vector $\mathbf{g} - 2\mathbf{f}$.
(a(ii))[3]
Petra states $|\mathbf{f}| > |\mathbf{g}|$. Show that this is wrong.
(b(i))[1]
Find $\overrightarrow{PB}$ expressed in terms of $\mathbf{a}$ and $\mathbf{b}$.
(b(ii))[2]
Find $\overrightarrow{OQ}$ in terms of $\mathbf{a}$ and $\mathbf{b}$, and simplify your answer completely.
(b(iii))[3]
Use vectors: from $QR = 2OQ$, show that $AR$ is parallel to $PB$.
Worked solution & mark scheme
This 10-mark question has a full step-by-step worked solution and mark scheme. One marking point: “The vector is $\begin{pmatrix}-7\\1\end{pmatrix}$.” …