VectorsAQA A-Level Maths: Topic test

1. 1
   The vectors $\mathbf{u}$ and $\mathbf{v}$ are given by $\mathbf{u}=8\mathbf{i}+15\mathbf{j}$ and $\mathbf{v}=-2\mathbf{i}+3\mathbf{j}$.

   (a)

   Which expression is $\mathbf{u}-2\mathbf{v}$?

   [1 mark]

   • A$4\mathbf{i}+21\mathbf{j}$
   • B$10\mathbf{i}+12\mathbf{j}$
   • C$12\mathbf{i}+9\mathbf{j}$
   • D$12\mathbf{i}+21\mathbf{j}$

   (b)

   What is the magnitude of $\mathbf{u}$?

   [1 mark]

   • A17
   • B23
   • C$\sqrt{23}$
   • D289

   (c)

   Find the angle that $\mathbf{u}$ makes with the positive $x$-direction, correct to 1 decimal place.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Relative to an origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}=-2\mathbf{i}+5\mathbf{j}$ and $\mathbf{b}=6\mathbf{i}-\mathbf{j}$.

   (a)

   Which expression is $\overrightarrow{AB}$?

   [1 mark]

   • A$-8\mathbf{i}+6\mathbf{j}$
   • B$8\mathbf{i}-6\mathbf{j}$
   • C$4\mathbf{i}+4\mathbf{j}$
   • D$8\mathbf{i}+6\mathbf{j}$

   (b)

   What is the distance $AB$?

   [1 mark]

   • A100
   • B14
   • C2
   • D10

   (c)

   The point $M$ is the midpoint of $AB$. Find the position vector of $M$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   In still air an aircraft would fly with velocity $(200\mathbf{i}+150\mathbf{j})$ km h$^{-1}$, where $\mathbf{i}$ and $\mathbf{j}$ are unit vectors due east and due north. There is a wind of velocity $(-40\mathbf{i}+30\mathbf{j})$ km h$^{-1}$, and the aircraft's actual velocity is the sum of the two.

   (a)

   Find the actual velocity of the aircraft and its speed, giving the speed to 3 significant figures.

   [3 marks]

   (b)

   Find the bearing on which the aircraft travels, to the nearest $0.1^\circ$, and its distance from its starting point after $2.5$ hours, to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Relative to a fixed origin $O$ at ground level, the unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ point east, north and vertically upwards, and distances are in metres. At time $t=0$ seconds a drone is at the point $A$ with position vector $(2\mathbf{i}-\mathbf{j}+3\mathbf{k})$. It then flies with constant velocity $(3\mathbf{i}+4\mathbf{j}+12\mathbf{k})$ m s$^{-1}$.

   (a)

   Find the speed of the drone, its position vector at time $t$, and its position vector at the moment its height above the ground is $63$ m.

   [6 marks]

   (b)

   The point $B$ has position vector $(14\mathbf{i}+15\mathbf{j}+51\mathbf{k})$. Show that the drone passes through $B$ and find the distance $AB$. Comment on how your answer relates to the speed of the drone.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   $OABC$ is a trapezium in which $OA$ is parallel to $CB$. $\overrightarrow{OA}=\mathbf{a}$, $\overrightarrow{OC}=\mathbf{c}$ and $\overrightarrow{CB}=3\mathbf{a}$.

   (a)

   Which expression is $\overrightarrow{AB}$?

   [1 mark]

   • A$2\mathbf{a}+\mathbf{c}$
   • B$3\mathbf{a}+\mathbf{c}$
   • C$4\mathbf{a}+\mathbf{c}$
   • D$\mathbf{c}-2\mathbf{a}$

   (b)

   The point $M$ is the midpoint of $OC$. Which expression is $\overrightarrow{MB}$?

   [1 mark]

   • A$3\mathbf{a}+\frac32\mathbf{c}$
   • B$\mathbf{a}+\frac12\mathbf{c}$
   • C$3\mathbf{a}+\frac12\mathbf{c}$
   • D$3\mathbf{a}-\frac12\mathbf{c}$

   (c)

   The point $N$ is the midpoint of $AB$. Find $\overrightarrow{ON}$ in terms of $\mathbf{a}$ and $\mathbf{c}$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The vector $\mathbf{w}$ is given by $\mathbf{w}=-9\mathbf{i}-12\mathbf{j}$.

   (a)

   What is $|\mathbf{w}|$?

   [1 mark]

   • A21
   • B3
   • C225
   • D15

   (b)

   What is the direction of $\mathbf{w}$, measured anticlockwise from the positive $x$-direction?

   [1 mark]

   • A$53.1^\circ$
   • B$233.1^\circ$
   • C$126.9^\circ$
   • D$306.9^\circ$

   (c)

   Find a vector in the same direction as $\mathbf{w}$ with magnitude $5$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A robot moves across a flat floor. The unit vectors $\mathbf{i}$ and $\mathbf{j}$ lie along two perpendicular walls, with the origin $O$ at their corner, and distances are in metres. At time $t=0$ seconds the robot is at the point with position vector $(2\mathbf{i}+9\mathbf{j})$ and it moves with constant velocity $(3\mathbf{i}-4\mathbf{j})$ m s$^{-1}$.

   (a)

   Find the position vector of the robot at time $t$ and the speed of the robot.

   [3 marks]

   (b)

   The robot crosses the wall along the $x$-axis. Find the time at which this happens and the distance of the robot from $O$ at that moment.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Relative to a fixed origin $O$, with $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ pointing east, north and vertically upwards (distances in metres), two drones $P$ and $Q$ have position vectors at time $t$ seconds of $\mathbf{r}_P=(1+2t)\mathbf{i}+(3-t)\mathbf{j}+(2+t)\mathbf{k}$ and $\mathbf{r}_Q=(11-3t)\mathbf{i}+(-3+2t)\mathbf{j}+(10-3t)\mathbf{k}$, for $0\le t\le4$.

   (a)

   Find the speed of each drone and show that the drones collide. State when and where this happens.

   [6 marks]

   (b)

   Find $\overrightarrow{PQ}$ at time $t$ and show that it is always parallel to a fixed vector. Hence find the distance between the drones when $t=0.5$, to 3 significant figures.

   [6 marks]

   Total for question 8: 12 marks

End of questions