GCSE Maths · Higher · Chapter 25: Vectors

Column vectors, vector arithmetic and vector proofs in geometry

Grades 4–9About 85 minutesNo calculator92 marks

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Before you start

Grades 4–5

You will need these in every question below.

A1
$a = (2 - 3)$ . Write $4 a$ as a column vector.
[1 mark]
Answer

A2
Work out the length of the vector $(512)$ .
[1 mark]
Answer
A3
$a = (14)$ and $b = (3 - 2)$ . Write $2 a + 3 b$ as a column vector.
[2 marks]
Answer
A4
$P$ is the point $(1, 2)$ and $Q$ is the point $(7, - 1)$ . Write $P Q$ as a column vector.
[1 mark]
Answer
A5
The point $A (2, 5)$ is translated by the vector $(3 - 4)$ . Write down the coordinates of its image.
[1 mark]
Answer

Examples, then your turn

Grades 5–6

Your teacher works through each example with you. Then try the one beside it on your own.

Example 1

$a = (3 - 1)$ and $b = (- 2 4)$ . Write $2 a - 3 b$ as a column vector.

Hint Multiply each vector first, then subtract the top numbers and the bottom numbers separately.

B1
$p = (25)$ and $q = (- 1 3)$ . Write $3 p + 2 q$ as a column vector.
[2 marks]
Answer

Example 2

$a = (41)$ and $b = (- 2 7)$ . $2 c + a = b$ . Find $c$ as a column vector.

Hint Rearrange as you would an equation:

2 c = b - a

B2
$a = (5 - 2)$ and $b = (18)$ . $3 c - a = b$ . Find $c$ as a column vector.
[3 marks]
Answer

Example 3

$O A B$ is a triangle. $O A = a$ and $O B = b$ . $M$ is the point on $A B$ with $A M : M B = 1 : 3$ . Find $O M$ in terms of $a$ and $b$ .

Not drawn accurately

Hint Go from

O

A

, then a quarter of the way from

A

B

A B = b - a

B3
$O A B$ is a triangle. $O A = a$ and $O B = b$ . $P$ is the point on $A B$ with $A P : P B = 3 : 2$ . Find $O P$ in terms of $a$ and $b$ . Simplify your answer.
[3 marks]
Not drawn accurately
Answer

Example 4

$P Q = (6 - 4)$ and $R S = (- 9 6)$ . Show that $P Q$ and $R S$ are parallel.

Hint Find a number that multiplies one vector to give the other.

B4
$K L = (4 - 10)$ and $M N = (- 6 15)$ . Show that $K L$ and $M N$ are parallel.
[2 marks]

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
The grid shows the vectors $a$ , $b$ and $c$ .
(a)
Write $a$ as a column vector.
[1 mark]
Answer
(b)
Write $b$ as a column vector.
[1 mark]
Answer
(c)
Write $c$ as a column vector.
[1 mark]
Answer
Total for C1: 3 marks

C2
$a = (5 - 1)$ and $b = (- 2 - 11)$ . Work out the length of the vector $2 a - b$ .
[3 marks]
Answer
C3
$p (23) + q (1 - 1) = (87)$ . Find the value of $p$ and the value of $q$ .
[3 marks]
Answer $p = \dots\dots$ $q = \dots\dots$
C4
$A$ is the point $(1, 2)$ , $B$ is $(4, 8)$ and $C$ is $(6, 12)$ . Use vectors to show that $A$ , $B$ and $C$ lie on a straight line.
[3 marks]
C5
$O A B C$ is a parallelogram. $O A = a$ and $O C = c$ . $M$ is the midpoint of $C B$ .
Not drawn accurately
(a)
Find $A C$ in terms of $a$ and $c$ .
[2 marks]
Answer
(b)
Find $O M$ in terms of $a$ and $c$ .
[2 marks]
Answer
Total for C5: 4 marks
C6
$O A B$ is a triangle. $O A = a$ and $O B = b$ . $P$ is the point on $O A$ with $O P : P A = 3 : 1$ , and $Q$ is the point on $O B$ with $O Q : QB = 3 : 1$ .
Show that $P Q$ is parallel to $A B$ , and find the ratio $P Q : A B$ .
[3 marks]
Not drawn accurately
C7
Each pair of vectors below is parallel.
(a)
$(k 6)$ is parallel to $(23)$ . Find $k$ .
[2 marks]
Answer
(b)
$(- 2 m)$ is parallel to $(12)$ . Find $m$ .
[2 marks]
Answer
Total for C7: 4 marks
C8
The point $A (- 1, 3)$ is translated by $(4 - 2)$ and then by $(- 3 5)$ .
(a)
Write the single vector that does both translations.
[1 mark]
Answer
(b)
Write down the coordinates of the final image of $A$ .
[1 mark]
Answer
Total for C8: 2 marks
C9
$A B C D E F$ is a regular hexagon with centre $O$ . $O A = a$ and $O B = b$ .
(a)
Find $A B$ in terms of $a$ and $b$ .
[1 mark]
Answer
(b)
Find $A D$ in terms of $a$ .
[2 marks]
Answer
(c)
Find $C E$ in terms of $a$ and $b$ .
[2 marks]
Answer
Total for C9: 5 marks

Exam-style questions

Grades 6–9

Answer every question in the space given. Show your working: most of the marks are for method.

D1
Sam says, " $6 a - 2 b$ is parallel to $3 a - 2 b$ , because $6 a - 2 b$ is $2$ lots of $3 a - 2 b$ ."
$a$ and $b$ are not parallel. Sam is wrong. Explain why.
[2 marks]
Answer

D2
$A B C D$ is a parallelogram. $A$ is the point $(1, 2)$ , $B$ is $(6, 4)$ and $D$ is $(8, - 5)$ .
Find the coordinates of $C$ .
[3 marks]
Answer
D3
$a$ and $b$ are vectors that are not parallel.
$O A = 2 a + b$ , $O B = a + 3 b$ , $O C = k a + 7 b$
The points $A$ , $B$ and $C$ lie on a straight line.
(a)
Find $A B$ in terms of $a$ and $b$ .
[1 mark]
Answer
(b)
Find the value of $k$ .
[3 marks]
Answer
(c)
Find the ratio $A B : B C$ .
[2 marks]
Answer
Total for D3: 6 marks
D4
$O A B$ is a triangle. $O A = a$ and $O B = b$ . $P$ is a point on $A B$ with $O P = \frac{1}{4} a + \frac{3}{4} b$ .
Find the ratio $A P : P B$ .
[3 marks]
Answer
D5
$p = (3 - 1)$ and $q = (- 2 4)$ . The vector $p + k q$ is parallel to $(12)$ .
Find the value of $k$ .
[3 marks]
Answer
D6
$O A B C$ is a trapezium. $O A = a$ , $O C = c$ and $C B = 2 a$ . $M$ is the midpoint of $A B$ .
Find $O M$ in terms of $a$ and $c$ . Simplify your answer.
[3 marks]
Not drawn accurately
Answer
D7
$A B = (5 - 12)$ and $B C = (- 8 6)$ .
Find the length of $A C$ . Give your answer as a surd in its simplest form.
[3 marks]
Answer
D8
The vector $(k k + 7)$ has length $13$ .
Find the two possible values of $k$ .
[3 marks]
Answer $k = \dots\dots$ or $k = \dots\dots$
D9
$a$ and $b$ are vectors that are not parallel.
$x a + 3 b + 2 (a - y b) = 7 a - 5 b$
Find the value of $x$ and the value of $y$ .
[3 marks]
Answer $x = \dots\dots$ $y = \dots\dots$
D10
The grid shows $O A = a$ and $O B = b$ .
(a)
Write $a - b$ as a column vector.
[2 marks]
Answer
(b)
$M$ is the midpoint of $A B$ . Write $O M$ as a column vector.
[2 marks]
Answer
Total for D10: 4 marks

Extension

Grade 9 and beyond

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
$O A B$ is a triangle. $O A = a$ and $O B = b$ . $M$ is the midpoint of $O B$ . $N$ is the point on $O A$ with $O N : N A = 1 : 2$ . The lines $A M$ and $B N$ meet at $X$ .
Find $O X$ in terms of $a$ and $b$ .
[4 marks]
Not drawn accurately
Hint 1 · What to try
Write $O X$ two ways: once going along $A M$ and once going along $B N$ .
Hint 2 · The first line
$O X = a + s (\frac{1}{2} b - a)$ and $O X = b + t (\frac{1}{3} a - b)$ .
Hint 3 · The full method
Match the $a$ parts and the $b$ parts: $1 - s = \frac{t}{3}$ and $\frac{s}{2} = 1 - t$ . These give $s = \frac{4}{5}$ .
Answer

E2
$O A B C$ is a parallelogram. $O A = a$ and $O C = c$ . $P$ is the point on $O A$ with $O P : P A = 1 : 2$ . The line $C P$ meets the diagonal $O B$ at $X$ .
Find the ratio $O X : X B$ .
[3 marks]
Hint 1 · What to try
Write $O X$ two ways: as a multiple of $O B$ , and by going from $O$ to $C$ and then part of the way along $C P$ .
Hint 2 · The first line
$O X = t (a + c)$ and $O X = c + s (\frac{1}{3} a - c)$ .
Hint 3 · The full method
Match the $a$ parts and the $c$ parts: $t = \frac{s}{3}$ and $t = 1 - s$ . These give $t = \frac{1}{4}$ .
Answer
E3
$A B C D$ is a trapezium with $A B$ parallel to $D C$ and $A B = 2 D C$ . The diagonals $A C$ and $B D$ meet at $X$ .
Use vectors to show that $A X : X C = 2 : 1$ .
[3 marks]
Hint 1 · What to try
Let $D C = p$ and $A D = d$ . Then $A B = 2 p$ .
Hint 2 · The first line
$A X = t (d + p)$ along $A C$ , and $A X = 2 p + s (d - 2 p)$ along $B D$ .
Hint 3 · The full method
Matching the parts gives $t = s$ and $t = 2 - 2 s$ , so $t = \frac{2}{3}$ .
E4
$a = (34)$ . The vector $b$ is parallel to $a$ but points the opposite way, and the length of $b$ is $15$ .
Write $b$ as a column vector.
[3 marks]
Hint 1 · What to try
Work out the length of $a$ first.
Hint 2 · The first line
$a$ has length $5$ , so $b$ is $3$ times as long.
Hint 3 · The full method
Opposite direction means a negative multiple: $b = - 3 a$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can add, subtract and multiply column vectors, find the length of a vector, and read vectors off a grid.
I can write a path round a shape in terms of given vectors, and use vectors to show that lines are parallel or that points lie on a straight line.

Answers and mark scheme

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