Relative to the origin $O$, the position vectors of points $A$, $B$ and $C$ are $\vec{OA} = \begin{pmatrix}0 \\ 5 \\ 2\end{pmatrix}$, $\vec{OB} = \begin{pmatrix}1 \\ 0 \\ 1\end{pmatrix}$ and $\vec{OC} = \begin{pmatrix}4 \\ -3 \\ -2\end{pmatrix}$. The midpoint of $AC$ is $M$ and $N$ is a point on $BC$, between $B$ and $C$, such that $BN = 2NC$.
(a)[3]
Determine the position vectors of $M$ and $N$.
(b)[2]
Determine a vector equation for the line passing through $M$ and $N$.
(c)[4]
Find the position vector of the point $Q$ where the line through $M$ and $N$ meets the line through $A$ and $B$.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “State $\overrightarrow{OM}=\begin{pmatrix}2\\1\\0\end{pmatrix}$” …