In the diagram, $OABCDEFG$ forms a cuboid, with $OA = 2$ units, $OC = 4$ units and $OG = 2$ units. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OC$ and $OG$ respectively. Point $M$ is the midpoint of $DF$. Point $N$ lies on $AB$ so that $AN = 3NB$.
(a)[3]
Write the vectors $\overrightarrow{OM}$ and $\overrightarrow{MN}$ in $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ notation.
(b)[2]
Find a vector equation of the line passing through $M$ and $N$.
(c)[4]
Show that the perpendicular distance from $O$ to the line through $M$ and $N$ is $\sqrt{\frac{53}{6}}$.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Obtain $\overrightarrow{OM}=\mathbf{i}+2\mathbf{j}+2\mathbf{k}$.” …