1. 1
   A games designer programs a character's movement on a grid using vectors. The character moves from point A to point B via vector $\begin{pmatrix}3\\-2\end{pmatrix}$, and then from point B to point C via vector $\begin{pmatrix}-5\\4\end{pmatrix}$.

   (a)

   Find the vector representing the character's movement from A to C.

   [1 mark]

   • A$\begin{pmatrix}-8\\6\end{pmatrix}$
   • B$\begin{pmatrix}8\\-6\end{pmatrix}$
   • C$\begin{pmatrix}2\\-2\end{pmatrix}$
   • D$\begin{pmatrix}-2\\2\end{pmatrix}$

   (b)

   Find the vector representing the movement from C back to A.

   [1 mark]

   • A$\begin{pmatrix}-2\\2\end{pmatrix}$
   • B$\begin{pmatrix}2\\-2\end{pmatrix}$
   • C$\begin{pmatrix}8\\-6\end{pmatrix}$
   • D$\begin{pmatrix}-8\\6\end{pmatrix}$

   (c)

   Find $2 \times$ the vector from A to B.

   [1 mark]

   • A$\begin{pmatrix}-6\\4\end{pmatrix}$
   • B$\begin{pmatrix}3\\-2\end{pmatrix}$
   • C$\begin{pmatrix}6\\-4\end{pmatrix}$
   • D$\begin{pmatrix}1.5\\-1\end{pmatrix}$

   Total for question 1: 3 marks

2. 2
   A hiker walks from a base camp to a lookout point along vector $\mathbf{p}=\begin{pmatrix}4\\3\end{pmatrix}$ kilometres, then from the lookout point to a summit along vector $\mathbf{q}=\begin{pmatrix}-1\\6\end{pmatrix}$ kilometres.

   (a)

   Find the vector $\mathbf{p}+\mathbf{q}$ representing the hiker's overall displacement from the base camp to the summit.

   [2 marks]

   (b)

   Find the magnitude of the hiker's overall displacement from the base camp to the summit, giving your answer to 1 decimal place.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   In triangle OAB, $\vec{OA}=\mathbf{a}$ and $\vec{OB}=\mathbf{b}$. Point M is the midpoint of AB.

   (a)

   Show that $\vec{OM}=\frac{1}{2}\mathbf{a}+\frac{1}{2}\mathbf{b}$.

   [3 marks]

   (b)

   Point N lies on OB such that $\vec{ON}=\frac{1}{2}\mathbf{b}$. Prove that MN is parallel to OA.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A drone delivery company models flight paths using vectors, where each unit represents 100 metres. A drone flies from the depot to Warehouse A via vector $\mathbf{a}=\begin{pmatrix}8\\6\end{pmatrix}$, then from Warehouse A to Warehouse B via vector $\mathbf{b}=\begin{pmatrix}-3\\4\end{pmatrix}$, then from Warehouse B back to the depot via vector $\mathbf{c}$.

   (a)

   Given that the drone returns exactly to the depot at the end of its route, find vector $\mathbf{c}$, the flight from Warehouse B back to the depot.

   [4 marks]

   (b)

   Find the total distance flown by the drone for the entire round trip, to the nearest metre.

   [4 marks]

   (c)

   The company wants to place a relay tower at the midpoint of the direct straight-line path from the depot to Warehouse B. Taking the depot as the origin, find the position vector of the relay tower, then find the direct distance from the depot to the relay tower, giving your answer to 1 decimal place.

   [5 marks]

   Total for question 4: 13 marks

End of questions