The quadrilateral $ABCD$ is a trapezium with $AB \parallel DC$. Relative to the origin $O$, the position vectors of $A$, $B$ and $C$ are $\overrightarrow{OA} = -\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$, $\overrightarrow{OB} = \mathbf{i} + 3\mathbf{j} + \mathbf{k}$ and $\overrightarrow{OC} = 2\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}$.
(a)[3]
Given that $\overrightarrow{DC} = 3\overrightarrow{AB}$, find the position vector of $D$ using this relation.
(b)[1]
State a vector equation for the line that passes through $A$ and $B$.
(c)[5]
Find the distance between the parallel sides, and hence find the area of the trapezium.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “State or indicate $\vec{AB}=2\mathbf{i}+\mathbf{j}-2\mathbf{k}$” …