1. 1
   The points $A$, $B$ and $C$ have coordinates $A(2,-1,3)$, $B(5,3,3)$ and $C(-1,0,7)$.

   (a)

   Find $\overrightarrow{AB}$.

   [1 mark]

   • A$\begin{pmatrix}-3\\-4\\0\end{pmatrix}$
   • B$\begin{pmatrix}3\\4\\0\end{pmatrix}$
   • C$\begin{pmatrix}7\\2\\6\end{pmatrix}$
   • D$\begin{pmatrix}3\\2\\0\end{pmatrix}$

   (b)

   Find the distance $AB$.

   [1 mark]

   • A$5$
   • B$25$
   • C$7$
   • D$\sqrt{7}$

   (c)

   Find a unit vector in the direction of $\overrightarrow{AC}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The vectors $\mathbf{p}$ and $\mathbf{q}$ are given by $\mathbf{p}=2\mathbf{i}-\mathbf{j}+2\mathbf{k}$ and $\mathbf{q}=\mathbf{i}+3\mathbf{j}-\mathbf{k}$.

   (a)

   Find $3\mathbf{p}-2\mathbf{q}$.

   [1 mark]

   • A$8\mathbf{i}+3\mathbf{j}+4\mathbf{k}$
   • B$4\mathbf{i}+3\mathbf{j}+8\mathbf{k}$
   • C$4\mathbf{i}-9\mathbf{j}+8\mathbf{k}$
   • D$4\mathbf{i}-9\mathbf{j}+4\mathbf{k}$

   (b)

   Which vector has magnitude 6 and the same direction as $\mathbf{p}$?

   [1 mark]

   • A$-4\mathbf{i}+2\mathbf{j}-4\mathbf{k}$
   • B$12\mathbf{i}-6\mathbf{j}+12\mathbf{k}$
   • C$6\mathbf{i}-3\mathbf{j}+6\mathbf{k}$
   • D$4\mathbf{i}-2\mathbf{j}+4\mathbf{k}$

   (c)

   The vector $m\mathbf{i}+2\mathbf{j}+n\mathbf{k}$ is parallel to $\mathbf{p}$. Find the value of $m$ and the value of $n$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   $P$, $Q$, $R$ and $S$ are four points in space, not necessarily in one plane, with position vectors $\mathbf{p}$, $\mathbf{q}$, $\mathbf{r}$ and $\mathbf{s}$ relative to an origin $O$. The points $K$, $L$, $M$ and $N$ are the midpoints of $PQ$, $QR$, $RS$ and $SP$ respectively.

   (a)

   Show that $\overrightarrow{KL}=\frac12(\mathbf{r}-\mathbf{p})$.

   [3 marks]

   (b)

   Hence prove that $KLMN$ is a parallelogram.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A drone takes off from the point $O$ on flat ground. Distances are in metres, with $\mathbf{i}$ pointing east, $\mathbf{j}$ pointing north and $\mathbf{k}$ vertically upwards, and $O$ as the origin. The drone flies in a straight line from $O$ to $A(30,60,60)$, then in a straight line from $A$ to $B(150,-100,60)$. Its battery allows a total flight distance of at most 450 m. A camera is fixed at the point $F(90,30,90)$.

   (a)

   (i) Find $\overrightarrow{AB}$ and the distance from $A$ to $B$.
   (ii) After reaching $B$, the drone is to fly in a straight line back to $O$. Determine whether its battery allows this.

   [6 marks]

   (b)

   The point $D$ lies on the line segment $AB$ such that $AD:DB=1:3$.
   (i) Find the coordinates of $D$.
   (ii) Show that $O$, $D$ and $F$ are collinear, and find the ratio $OD:DF$.

   [6 marks]

   Total for question 4: 12 marks

End of questions