GCSE Maths · Higher · Chapter 25: Vectors · Lesson 1

Properties of vectors

Grades 5–8About 50 minutesNo calculator40 marks

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NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Reflect $(5, - 6)$ in the $x$ -axis.
[1 mark]
Answer
A2
From chapter 16
Work out $2 7^{\frac{2}{3}}$ .
[1 mark]
Answer
A3
From chapter 22
Find $a$ from $a^{2} = b^{2} + c^{2} - 2 b c cos A$ with $b = 5$ , $c = 8$ , $A = 6 0^{\circ}$ . ( $cos 6 0^{\circ} = 0.5$ )
[1 mark]
Answer
A4
From chapter 1
Write $72$ as a product of its prime factors.
[1 mark]
Answer

Example, then your turn

Grades 5–7

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

Example 1

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

Hint Multiply each vector first, then subtract the tops and subtract the bottoms.

B1
$p = (- 2 5)$ and $q = (3 - 4)$ . Write $4 p + 3 q$ as a column vector.
[2 marks]
Answer

Example 2

$a = (27)$ and $b = (1 - 1)$ . Work out the length of $a + 3 b$ .

Hint Find the column vector first. Its length is the hypotenuse of a right-angled triangle with the top and bottom numbers as the other two sides.

B2
$u = (104)$ and $w = (34)$ . Work out the length of $u + 2 w$ .
[3 marks]
Answer

Practice

Grades 5–7

C1
A point $A$ is translated by $(4 - 7)$ . Its image is $B (1, 3)$ . Write down the coordinates of $A$ .
[2 marks]
Answer
C2
Simplify $3 (2 a - b) - 2 (a - 4 b)$ .
[2 marks]
Answer

C3
The grid shows the vectors $u$ and $v$ . Write $u - 2 v$ as a column vector.
[2 marks]
Answer
C4
$(6 m)$ is parallel to $(- 4 10)$ . Work out the value of $m$ .
[2 marks]
Answer
C5
$a$ and $b$ are not parallel. $(p + 2) a + 3 b = 7 a + (q - 1) b$ . Work out $p$ and $q$ .
[2 marks]
Answer $p = \dots\dots$ $q = \dots\dots$
C6
$u + 2 v = (94)$ and $u - v = (3 - 5)$ . Write $u$ and $v$ as column vectors.
[3 marks]
Answer $u = \dots\dots$ $v = \dots\dots$

Exam-style questions

Grades 6–8

D1
Tom says, " $(3 - 4)$ and $(- 6 8)$ are not parallel, because one is not a positive multiple of the other."
Explain why Tom is wrong.
[2 marks]

D2
An aircraft flies with velocity $(12050)$ km/h in still air. The wind adds $(- 24 22)$ km/h. The first number is east and the second is north.
(a)
Write the aircraft's resultant velocity as a column vector.
[1 mark]
Answer
(b)
Work out the aircraft's speed.
[2 marks]
Answer $\dots\dots\dots$ km/h
(c)
The aircraft starts at the point $(10, 20)$ , in km. Work out where it is after $30$ minutes.
[2 marks]
Answer
Total for D2: 5 marks
D3
$A$ is the point $(1, 3)$ , $B$ is $(5, 1)$ and $C$ is $(15, - 4)$ .
(a)
Write $A B$ and $B C$ as column vectors.
[2 marks]
Answer $A B = \dots\dots$ $B C = \dots\dots$
(b)
Use your answers to explain why $A$ , $B$ and $C$ lie on a straight line.
[2 marks]
(c)
Work out the ratio $A B : B C$ .
[1 mark]
Answer
(d)
Work out the length of $A C$ . Give your answer as a surd in its simplest form.
[2 marks]
Answer
Total for D3: 7 marks

Extension

Grade 9 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
$p = (- 6 - 4)$ and $q = (2 - 2)$ . The vector $p + k q$ has length $10$ . Work out the two possible values of $k$ .
[4 marks]
Hint 1 · What to try
Write $p + k q$ as one column vector with $k$ in it.
Hint 2 · The first line
$(2 k - 6 - 2 k - 4)$ has length $10$ , so $(2 k - 6)^{2} + (- 2 k - 4)^{2} = 100$ .
Hint 3 · The full method
This gives $8 k^{2} - 8 k - 48 = 0$ , so $k^{2} - k - 6 = 0$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can add, subtract and multiply column vectors.
I can work out the length of a vector.
I can decide whether two vectors are parallel.

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