From the diagram, $OABCD$ is a solid shape with $OA = OB = 4$ units and $OD = 3$ units. The edge $OD$ is vertical, $DC$ runs parallel to $OB$, and $DC = 1$ unit. The base $OAB$ is horizontal, while $ngle AOB = 90^ 07$. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OB$ and $OD$ respectively. $M$ is the midpoint of $AB$, and $N$ lies on $BC$ so that $CN = 2NB$.
(a)[4]
Write $\overrightarrow{MD}$ and $\overrightarrow{ON}$ in terms of $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$.
(b)[3]
Calculate the angle in degrees between the directions of $\overrightarrow{MD}$ and $\overrightarrow{ON}$.
(c)[4]
Show that the perpendicular distance from $M$ to $ON$ is $\sqrt{\dfrac{22}{5}}$.
Worked solution & mark scheme
This 11-mark question has a full step-by-step worked solution and mark scheme. One marking point: “State that $\overrightarrow{OM} = 2\mathbf{i} + 2\mathbf{j}$” …