Vectors in two and three dimensionsEdexcel International A Level Maths: Subtopic test

1. 1
   The points $A$ and $B$ have position vectors $\mathbf{a}=2\mathbf{i}-\mathbf{j}+3\mathbf{k}$ and $\mathbf{b}=5\mathbf{i}+3\mathbf{j}-\mathbf{k}$ relative to the origin $O$.

   (a)

   Find $\overrightarrow{AB}$.

   [1 mark]

   • A$-3\mathbf{i}-4\mathbf{j}+4\mathbf{k}$
   • B$7\mathbf{i}+2\mathbf{j}+2\mathbf{k}$
   • C$3\mathbf{i}+2\mathbf{j}-4\mathbf{k}$
   • D$3\mathbf{i}+4\mathbf{j}-4\mathbf{k}$

   (b)

   Find the exact value of $|\overrightarrow{AB}|$.

   [1 mark]

   • A$41$
   • B$\sqrt{29}$
   • C$\sqrt{41}$
   • D$3$

   (c)

   Find a unit vector in the direction of $\overrightarrow{AB}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The points $A$ and $B$ have coordinates $(-1,2,3)$ and $(3,-2,5)$ relative to the origin $O$, and $M$ is the midpoint of $AB$.

   (a)

   Find the position vector of $M$.

   [1 mark]

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

   (b)

   Find the distance $AB$.

   [1 mark]

   • A$6$
   • B$36$
   • C$\sqrt{28}$
   • D$10$

   (c)

   The point $C$ is such that $B$ is the midpoint of $AC$. Find the position vector of $C$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Triangle $ABC$ has vertices $A(1,2,-1)$, $B(4,-2,3)$ and $C(5,5,3)$, where the coordinates are in metres. A calculator may be used.

   (a)

   Show that triangle $ABC$ is isosceles.

   [3 marks]

   (b)

   The point $M$ is the midpoint of $BC$. Find the position vector of $M$, and hence find the exact area of triangle $ABC$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   $OABC$ is a parallelogram, where $O$ is the origin, $\overrightarrow{OA}=4\mathbf{i}+\mathbf{j}+2\mathbf{k}$ and $\overrightarrow{OC}=-\mathbf{i}+3\mathbf{j}+\mathbf{k}$. The point $M$ is the midpoint of $AB$.

   (a)

   (i) Find $\overrightarrow{OB}$.
   (ii) Find the position vector of $M$.
   (iii) Find the exact length $OM$.

   [6 marks]

   (b)

   The point $D$ lies on $OM$ extended so that $\overrightarrow{OD}$ is in the same direction as $\overrightarrow{OM}$ and $|OD|=2|OM|$. Find the position vector of $D$, and hence show that $OADB$ is a parallelogram.

   [6 marks]

   Total for question 4: 12 marks

End of questions