G24, G25Geometry & MeasuresFoundation & HigherGrades 4–9

Vectors

Q: How do I write vectors by hand?

Underline the letter: a. Bold type is only used in print.

Q: What is the magnitude of a vector?

Its length, found with Pythagoras: | pmatrix 3 4 pmatrix | = 3^2 + 4^2 = 5.

Q: Is AB the same as BA?

No. They have the same length but opposite directions: BA = - AB.

A vector describes a movement with both a size and a direction. It can be written as a column vector such as $(3 - 2)$ , as a bold letter $a$ , or as $A B$ . Vector arithmetic is simple, and it powers the hardest geometry proofs on the Higher paper.

50 min AQA GCSE Maths

The big picture

A vector is a journey, not a place. $A B$ means "the journey from $A$ to $B$ " — and any route from $A$ to $B$ gives the same total. That is why you can find an unknown vector by going the long way round through points you know.

Two vectors are parallel when one is a multiple of the other. Showing that $P Q = 3 R S$ proves the lines are parallel; showing $P Q = k QR$ proves $P$ , $Q$ and $R$ lie on a straight line. Those two facts are the core of every vector proof.

What you'll learn

Use vector notation and column vectors
Add, subtract and multiply vectors by a scalar
Use vectors to describe routes around a shape
Prove results about parallel lines and collinear points using vectors (Higher)

Column vectors and arithmetic

In $(x y)$ , the top number is the movement right (negative means left) and the bottom is the movement up (negative means down).

Add and subtract component by component. Multiplying by a scalar multiplies both components: $3 (2 - 1) = (6 - 3)$ . The vector $- a$ has the same length as $a$ but the opposite direction.

2 a = (82)

and

3 b = (- 6 9)

2Subtract:

(8 - (- 6) 2 - 9)

Answer

(14 - 7)

Your turnSame method, new numbers — try it before moving on.

$p = (5 - 2)$ and $q = (14)$ . Find $p + 2 q$ .

Your answer

Type the column vector as (top, bottom), e.g. (3, -2).

Your turnSame method, new numbers — try it before moving on.

$P$ is $(- 2, 3)$ and $Q$ is $(4, - 1)$ . Write $QP$ as a column vector.

Your answer

Type the column vector as (top, bottom), e.g. (3, -2).

Vectors around shapes

To find an unknown vector, travel along known vectors from the start to the end. Going along a vector backwards changes its sign: $B A = - A B$ .

For a point part-way along a line, use fractions of the vector: if $M$ is the midpoint of $A B$ , then $A M = \frac{1}{2} A B$ .

O A = a

O B = b

and

M

is the midpoint of

A B

A B = A O + O B = - a + b

O M = O A + \frac{1}{2} A B = a + \frac{1}{2} (b - a)

3Simplify:

\frac{1}{2} a + \frac{1}{2} b

Answer

A B = b - a

;

O M = \frac{1}{2} (a + b)

Tip — A ratio $1 : 3$ splits a line into 4 parts, so $A N$ is $\frac{1}{4}$ of $A C$ — not $\frac{1}{3}$ .

Parallel vectors and proofs (Higher)

Vectors are parallel if one is a scalar multiple of the other: $(6 - 9) = 3 (2 - 3)$ , and $4 a - 2 b = 2 (2 a - b)$ .

To prove three points $P$ , $Q$ , $R$ are on a straight line (collinear), show $P Q$ is a multiple of $QR$ (they are parallel) $and$ state that they share the point $Q$ .

P Q = 2 (a + 3 b) = 2 QR

2So

P Q

is parallel to

QR

3They share the point

Q

, so

P

Q

and

R

lie on a straight line.

AnswerCollinear, since

P Q = 2 QR

with common point

Q

Your turnSame method, new numbers — try it before moving on.

Which value of $k$ makes $(k 10)$ parallel to $(35)$ ?

Your answer

Why the common point matters

Parallel alone is not enough for collinearity: two train tracks are parallel but never meet. The shared point is what forces the two segments onto the same line — which is why the examiner wants it stated.

Think like an examiner

Write the route you are taking, e.g. $O M = O A + A M$ , before substituting — it earns method marks.
Simplify fully: collect the $a$ terms and the $b$ terms.
In proofs, a final sentence using the words "parallel" and "common point" is needed for the last mark.

Vector facts

A B = A O + O B = b - a

Travel via known points.

B A = - A B

Reverse direction, change sign.

u = k v \Rightarrow u ∥ v

Parallel vectors.

Watch out for these

A B = a - b

when

O A = a

O B = b

A B = - a + b = b - a

Writing a column vector as coordinates

(3, - 2)

in working.

Use

(3 - 2)

For

A N : N C = 1 : 3

, using

A N = \frac{1}{3} A C

There are 4 parts:

\frac{1}{4} A C

Concluding collinear from parallel vectors alone.

Also state the common point.

Stretch yourself

$O A B$ is a triangle with $O A = a$ and $O B = b$ . $P$ is the midpoint of $O A$ and $Q$ is the midpoint of $O B$ . Prove that $P Q$ is parallel to $A B$ and half its length.

Hint — Find $P Q$ via $O$ , and compare it with $A B$ .

Questions students ask

How do I write vectors by hand?

Underline the letter:

\underline{a}

. Bold type is only used in print.

What is the magnitude of a vector?

Its length, found with Pythagoras:

(34) = 3^{2} + 4^{2} = 5

A B

the same as

B A

No. They have the same length but opposite directions:

B A = - A B

Key takeaways

Column vectors: top is right/left, bottom is up/down.
Add, subtract and scale component by component.
Find unknown vectors by travelling via known points; reversing changes the sign.
Use fractions of vectors for points dividing lines in a ratio.
Parallel means a scalar multiple; collinear also needs a common point.

Exam-style questions

Practise it the way it is examined

3 original questions · 9 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 42 marks

$a = (3 - 1)$ and $b = (- 2 5)$ .

Work out $3 a + b$ as a column vector.[2]

Question 2Grade 63 marks

$A B C D E F$ is a regular hexagon with centre $O$ . $O A = a$ and $O B = b$ .

Write $A B$ and $A C$ in terms of $a$ and $b$ .[3]

Question 3Grade 84 marks

$O A B$ is a triangle. $O A = a$ and $O B = b$ . $M$ is the midpoint of $A B$ . $N$ is the point on $O B$ such that $O N : N B = 2 : 1$ . $X$ is the point such that $O X = 2 a$ .

Prove that $X$ , $M$ and $N$ lie on a straight line.[4]

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Vectors

Column vectors and arithmetic

Vectors around shapes

Parallel vectors and proofs (Higher)

Practise it the way it is examined

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