Vectors in problem solvingEdexcel A-Level Maths: Subtopic test

1. 1
   Relative to an origin $O$, the points $A$, $B$ and $C$ have position vectors $\mathbf{a}=\mathbf{i}+2\mathbf{j}$, $\mathbf{b}=5\mathbf{i}+3\mathbf{j}$ and $\mathbf{c}=8\mathbf{i}+7\mathbf{j}$. The quadrilateral $ABCD$ is a parallelogram.

   (a)

   Find the position vector of $D$.

   [1 mark]

   • A$9\mathbf{i}+9\mathbf{j}$
   • B$4\mathbf{i}+6\mathbf{j}$
   • C$12\mathbf{i}+8\mathbf{j}$
   • D$-2\mathbf{i}-2\mathbf{j}$

   (b)

   Find the exact distance $AC$.

   [1 mark]

   • A$12$
   • B$74$
   • C$\sqrt{113}$
   • D$\sqrt{74}$

   (c)

   Find the position vector of the point where the diagonals $AC$ and $BD$ meet.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Two forces $\mathbf{F}_1=(5\mathbf{i}+2\mathbf{j})$ N and $\mathbf{F}_2=(-3\mathbf{i}+7\mathbf{j})$ N act on a particle, where $\mathbf{i}$ and $\mathbf{j}$ are unit vectors due east and due north. A third force $\mathbf{F}_3$ also acts, and the particle is in equilibrium.

   (a)

   Find the resultant of $\mathbf{F}_1$ and $\mathbf{F}_2$.

   [1 mark]

   • A$(8\mathbf{i}-5\mathbf{j})$ N
   • B$(2\mathbf{i}+5\mathbf{j})$ N
   • C$(2\mathbf{i}+9\mathbf{j})$ N
   • D$(8\mathbf{i}+9\mathbf{j})$ N

   (b)

   Find $\mathbf{F}_3$.

   [1 mark]

   • A$(-2\mathbf{i}-9\mathbf{j})$ N
   • B$(2\mathbf{i}+9\mathbf{j})$ N
   • C$(2\mathbf{i}-9\mathbf{j})$ N
   • D$(-8\mathbf{i}+5\mathbf{j})$ N

   (c)

   Find the angle that the resultant of $\mathbf{F}_1$ and $\mathbf{F}_2$ makes with the direction of $\mathbf{i}$, giving your answer to 1 decimal place.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   At time $t=0$ a ship leaves a point with position vector $(2\mathbf{i}-5\mathbf{j})$ km relative to a port $O$, and then moves with constant velocity $(3\mathbf{i}+4\mathbf{j})$ km h$^{-1}$, where $\mathbf{i}$ and $\mathbf{j}$ are unit vectors due east and due north.

   (a)

   Find the speed of the ship and its position vector after $t$ hours.

   [3 marks]

   (b)

   The ship is due east of $O$ at time $T$ hours. Find $T$ and the distance of the ship from $O$ at this time.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Relative to an origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}=6\mathbf{i}+2\mathbf{j}$ and $\mathbf{b}=-3\mathbf{i}+11\mathbf{j}$. The point $P$ lies on $AB$ such that $AP:PB=1:2$.

   (a)

   (i) Find $\overrightarrow{AB}$.
   (ii) Find the position vector of $P$.
   The point $R$ has position vector $9\mathbf{i}+15\mathbf{j}$.
   (iii) Show that $O$, $P$ and $R$ lie on a straight line.
   (iv) Find the exact distance $OP$.

   [6 marks]

   (b)

   The point $C$ is such that $OACB$ is a parallelogram.
   (i) Find the position vector of $C$.
   (ii) Show that the midpoint $M$ of $AB$ lies on $OC$.
   (iii) Show that $OACB$ is not a rectangle.

   [6 marks]

   Total for question 4: 12 marks

End of questions