Mathematics 9709 · AS & A Level · Coordinate geometry
Coordinate geometry — practice question
The figure represents a three-dimensional solid $OABCDEFG$. The base $OABC$ and the top face $DEFG$ are matching horizontal rectangles. The parallelograms $OAED$ and $CBFG$ lie in vertical planes. Points $P$ and $Q$ are the mid-points of $OD$ and $GF$ respectively. Unit vectors $\mathbf{i}$ and $\mathbf{j}$ are parallel to $\overrightarrow{OA}$ and $\overrightarrow{OC}$ respectively and the unit vector $\mathbf{k}$ is vertically upwards. The position vectors of $A$, $C$ and $D$ are given by $\overrightarrow{OA} = 6\mathbf{i}$, $\overrightarrow{OC} = 8\mathbf{j}$ and $\overrightarrow{OD} = 2\mathbf{i} + 10\mathbf{k}$.
(i)[4]
Express each of the vectors $\overrightarrow{PB}$ and $\overrightarrow{PQ}$ in terms of $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$.
(ii)[2]
Determine whether $P$ is nearer to $Q$ or to $B$.
(iii)[3]
Use a scalar product in order to find angle $BPQ$.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Vector $\vec{PB}=5\mathbf{i}+8\mathbf{j}-5\mathbf{k}$ is correct” …