Pure: VectorsEdexcel A-Level Maths: Topic test

1. 1
   The vectors $\mathbf{u}$ and $\mathbf{w}$ are given by $\mathbf{u}=7\mathbf{i}-24\mathbf{j}$ and $\mathbf{w}=-3\mathbf{i}+\mathbf{j}$.

   (a)

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

   [1 mark]

   • A$17$
   • B$31$
   • C$25$
   • D$625$

   (b)

   Which vector is the unit vector in the direction of $\mathbf{u}$?

   [1 mark]

   • A$\dfrac{1}{25}(7\mathbf{i}-24\mathbf{j})$
   • B$\dfrac{1}{25}(24\mathbf{i}-7\mathbf{j})$
   • C$\dfrac{1}{31}(7\mathbf{i}-24\mathbf{j})$
   • D$\dfrac{1}{25}(-7\mathbf{i}+24\mathbf{j})$

   (c)

   Find the value of $\lambda$ for which $\mathbf{u}+\lambda\mathbf{w}$ is parallel to $\mathbf{j}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Relative to an origin $O$, the points $P$ and $Q$ have position vectors $\mathbf{p}=-3\mathbf{i}+5\mathbf{j}$ and $\mathbf{q}=9\mathbf{i}-4\mathbf{j}$.

   (a)

   What is $\overrightarrow{PQ}$?

   [1 mark]

   • A$-12\mathbf{i}+9\mathbf{j}$
   • B$6\mathbf{i}+\mathbf{j}$
   • C$12\mathbf{i}+9\mathbf{j}$
   • D$12\mathbf{i}-9\mathbf{j}$

   (b)

   What is the distance $PQ$?

   [1 mark]

   • A$21$
   • B$15$
   • C$225$
   • D$3$

   (c)

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

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Relative to a fixed origin $O$, the points $A$, $B$ and $C$ have coordinates $A(2,1,-3)$, $B(8,4,3)$ and $C(3,5,5)$.

   (a)

   Find the distance $AB$.

   [3 marks]

   (b)

   Show that triangle $ABC$ is isosceles, and find its perimeter, giving your answer to $3$ significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In three dimensions, the points $P$, $Q$ and $R$ have coordinates $P(1,-2,4)$, $Q(5,0,10)$ and $R(8,3,7)$. The quadrilateral $PQRS$ is a parallelogram.

   (a)

   Find the position vector of $S$, and find the exact perimeter of $PQRS$.

   [6 marks]

   (b)

   The diagonals of $PQRS$ are $PR$ and $QS$. Show that the diagonals bisect each other, and that they have equal length. Hence state, with a reason, the special name of this parallelogram.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A boat has velocity $(6\mathbf{i}-8\mathbf{j})$ km h$^{-1}$ in still water, where $\mathbf{i}$ is a unit vector due east and $\mathbf{j}$ is a unit vector due north.

   (a)

   What is the speed of the boat in still water?

   [1 mark]

   • A$2$ km h$^{-1}$
   • B$10$ km h$^{-1}$
   • C$14$ km h$^{-1}$
   • D$100$ km h$^{-1}$

   (b)

   In which direction is the boat travelling?

   [1 mark]

   • A$36.9^\circ$ south of east
   • B$53.1^\circ$ north of east
   • C$36.9^\circ$ north of east
   • D$53.1^\circ$ south of east

   (c)

   A current of velocity $(-2\mathbf{i}+3\mathbf{j})$ km h$^{-1}$ also acts on the boat. Find the resultant velocity of the boat and its speed, to $3$ significant figures.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   In three dimensions, the points $A$ and $B$ have coordinates $A(1,-2,5)$ and $B(7,2,-7)$.

   (a)

   What is $\overrightarrow{AB}$?

   [1 mark]

   • A$6\mathbf{i}+4\mathbf{j}-12\mathbf{k}$
   • B$8\mathbf{i}-2\mathbf{k}$
   • C$-6\mathbf{i}-4\mathbf{j}+12\mathbf{k}$
   • D$6\mathbf{i}+4\mathbf{j}+12\mathbf{k}$

   (b)

   What is the distance $AB$?

   [1 mark]

   • A$196$
   • B$22$
   • C$14$
   • D$\sqrt{22}$

   (c)

   The point $C$ is such that $\overrightarrow{BC}=2\overrightarrow{AB}$. Find the coordinates of $C$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Three forces $\mathbf{F}_1=(2\mathbf{i}-5\mathbf{j})$ N, $\mathbf{F}_2=(p\mathbf{i}+3\mathbf{j})$ N and $\mathbf{F}_3=(-6\mathbf{i}+q\mathbf{j})$ N act on a particle of mass $2$ kg, where $p$ and $q$ are constants. The particle is in equilibrium.

   (a)

   Find the values of $p$ and $q$.

   [3 marks]

   (b)

   The force $\mathbf{F}_3$ is removed. Find the magnitude of the acceleration of the particle, and the angle its direction makes with $\mathbf{i}$, giving both answers to $3$ significant figures.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Relative to an origin $O$ at ground level, with $\mathbf{i}$ east, $\mathbf{j}$ north and $\mathbf{k}$ vertically upwards, the position vectors, in km, of two aircraft $A$ and $B$ at time $t$ minutes are $\mathbf{r}_A=(4\mathbf{i}-6\mathbf{j}+3\mathbf{k})+t(2\mathbf{i}+3\mathbf{j}+\mathbf{k})$ and $\mathbf{r}_B=(20\mathbf{i}-14\mathbf{j}+3\mathbf{k})+t(-2\mathbf{i}+5\mathbf{j}+\mathbf{k})$, for $t\geqslant0$.

   (a)

   Find the speed of each aircraft and the distance between them when $t=0$. State, with a reason, which aircraft is faster.

   [6 marks]

   (b)

   Show that the two aircraft collide, and find the time and the position vector of the collision.

   [6 marks]

   Total for question 8: 12 marks

End of questions