In the diagram, $OABCDEFG$ is a cuboid with $OA = 2$ units, $OC = 3$ units and $OD = 2$ units. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OC$ and $OD$, respectively. Point $M$ lies on $AB$ so that $MB = 2AM$. $N$ is the midpoint of $FG$.
(a)[3]
Express $\overrightarrow{OM}$ and $\overrightarrow{MN}$ using $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$.
(b)[2]
Find a vector equation of the line joining $M$ and $N$.
(c)[4]
Find the position vector of $P$, where $P$ is the foot of the perpendicular from $D$ to the line through $M$ and $N$.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Obtain the vector $\overrightarrow{OM} = 2\mathbf{i} + \mathbf{j}$” …