Position vectors and distanceEdexcel A-Level Maths: Subtopic test

1. 1
   Relative to an origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}=2\mathbf{i}+7\mathbf{j}$ and $\mathbf{b}=8\mathbf{i}-\mathbf{j}$.

   (a)

   Find $\overrightarrow{AB}$.

   [1 mark]

   • A$10\mathbf{i}+6\mathbf{j}$
   • B$-6\mathbf{i}+8\mathbf{j}$
   • C$6\mathbf{i}+6\mathbf{j}$
   • D$6\mathbf{i}-8\mathbf{j}$

   (b)

   Find the distance $AB$.

   [1 mark]

   • A$14$
   • B$10$
   • C$100$
   • D$\sqrt{136}$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Relative to an origin $O$, the points $P$, $Q$ and $R$ have position vectors $\mathbf{p}=\mathbf{i}+2\mathbf{j}$, $\mathbf{q}=5\mathbf{i}+5\mathbf{j}$ and $\mathbf{r}=13\mathbf{i}+11\mathbf{j}$.

   (a)

   Find $\overrightarrow{PQ}$.

   [1 mark]

   • A$6\mathbf{i}+7\mathbf{j}$
   • B$-4\mathbf{i}-3\mathbf{j}$
   • C$4\mathbf{i}+3\mathbf{j}$
   • D$4\mathbf{i}+7\mathbf{j}$

   (b)

   Which statement about the points $P$, $Q$ and $R$ is correct?

   [1 mark]

   • A$P$, $Q$ and $R$ lie on a straight line, with $PQ:QR=1:2$
   • B$P$, $Q$ and $R$ do not lie on a straight line, because $\overrightarrow{PQ}\neq\overrightarrow{QR}$
   • C$Q$ is the midpoint of $PR$
   • D$P$, $Q$ and $R$ lie on a straight line, with $PQ:QR=2:1$

   (c)

   Find the exact distance $PR$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Relative to an origin $O$, the points $A$, $B$ and $C$ are the vertices of a triangle, with position vectors $\mathbf{a}=\mathbf{i}+2\mathbf{j}$, $\mathbf{b}=7\mathbf{i}+4\mathbf{j}$ and $\mathbf{c}=3\mathbf{i}+8\mathbf{j}$.

   (a)

   Show that triangle $ABC$ is isosceles.

   [3 marks]

   (b)

   Let $M$ be the midpoint of $BC$. Find the area of triangle $ABC$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Three fence posts $A$, $B$ and $C$ stand in a field. Relative to a fixed origin $O$, their position vectors, in metres, are $\mathbf{a}=2\mathbf{i}+\mathbf{j}$, $\mathbf{b}=8\mathbf{i}+5\mathbf{j}$ and $\mathbf{c}=4\mathbf{j}$.

   (a)

   (i) Find $\overrightarrow{AB}$ and $\overrightarrow{AC}$.
   (ii) Show that angle $BAC$ is a right angle.
   (iii) Find the area of triangle $ABC$ in square metres.

   [6 marks]

   (b)

   A fourth post $D$ is placed so that $ABDC$ is a rectangle.
   (i) Find the position vector of $D$.
   (ii) Find the exact distance $OD$.
   (iii) Show that the midpoint of $AD$ is also the midpoint of $BC$.

   [6 marks]

   Total for question 4: 12 marks

End of questions