Expressing Vectors in Terms of a and b

Find any vector in a shape as a route of known sides: parallelograms, trapeziums, regular hexagons and points part way along a side.

In the exam for Cambridge O Level Maths 4024, Cambridge IGCSE Maths 0580, Edexcel Further Pure Maths 4PM1, Edexcel IGCSE Maths 4MA1, IGCSE Additional Maths 0606 · 62 real past-paper questions below

اقرأ هذا الدرس مشروحاً بالعربي ←

Routes Through a Shape

You can only use the vectors you are given. To find a new vector, walk from its start to its end along sides you know.

You walk	You write
along an arrow	plus that vector
against an arrow	minus that vector
a parallel side, same length	the same vector
a parallel side, twice as long	2 times the vector

Rule

Any route works: $A B = A O + O B$ . Pick the one through sides you know.

Parallelograms: Opposite Sides Match

In a parallelogram, opposite sides are equal and parallel, so they are the same vector.

Worked example

1
$O A B C : O A = a, O C = b$ — The parallelogram is OABC. We are given the two sides that start at O.
2
$A B = b$ — AB is opposite OC. Opposite sides of a parallelogram are equal and parallel, so AB is the same vector $b$ .
3
$O B = O A + A B$ — To get from O to B, go along OA, then along AB.
4
$= a + b$ — Put in $O A = a$ and $A B = b$ .
5
$A C = A O + O C$ — For the other diagonal, go from A back to O, then out to C.
6
$A O = - a$ — A to O goes against the arrow for OA, so it has a minus.
7
$A C = - a + b$ — Put in $A O = - a$ and $O C = b$ .
8
$= b - a$ — Write the positive term first. It is the same vector and easier to read.

Common mistake

Writing a − b for AC. Check the first step: from A to O goes against a, so it is −a.

Trapeziums: Parallel Sides Are Multiples

In a trapezium, one pair of sides is parallel but the lengths differ. The longer one is a number times the shorter one.

Worked example

1
$O A B C : O A = a, O C = b$ — The trapezium is OABC. We are given the two sides that start at O.
2
$A B = 2 b$ — Given: AB is parallel to OC and twice as long, so it is 2 lots of $b$ , pointing the same way.
3
$C B = C O + O A + A B$ — From C to B, go back to O, along to A, then along to B. All three sides are known.
4
$C O = - b$ — C to O goes against the arrow for OC, so it has a minus.
5
$= - b + a + 2 b$ — Put in CO, OA and AB.
6
$= a + b$ — Collect the $b$ terms: $- 1 + 2 = 1$ , so one $b$ .

Common mistake

Using b for AB. The side is parallel to OC but twice as long, so it is 2b.

Regular Hexagons

The 4-mark question uses a regular hexagon. Two facts do most of the work.

Fact	Vector
Opposite sides are parallel	DE is the reverse of AB
A long diagonal is parallel to two sides	twice as long as each of them

Worked example

1
$A B C D E F : A B = a, B C = b$ — A regular hexagon has equal sides and equal angles. We are given two sides that follow each other.
2
$A D = 2 b$ — AD is a long diagonal. It is parallel to BC and twice as long.
3
$D E = - a$ — DE is the side opposite AB. It is equal and parallel, but points the other way, so it has a minus.
4
$A E = A D + D E$ — To get from A to E, go along the long diagonal to D, then along DE.
5
$= 2 b + (- a)$ — Put in $A D = 2 b$ and $D E = - a$ .
6
$= 2 b - a$ — Adding $- a$ is the same as taking away $a$ .

Common mistake

Going round the edge one side at a time. It works, but the long diagonal is much quicker and has fewer signs to get wrong.

A Point Part of the Way Along a Side

When a point splits a side in a ratio, turn the ratio into a fraction of the whole side first.

Worked example

1
$O A = a, O B = b, A M : M B = 1 : 3$ — What we are given. M splits AB in the ratio 1 : 3.
2
$A B = A O + O B$ — First find the whole side. Go from A back to O, then out to B.
3
$= - a + b$ — A to O goes against OA, so it is $- a$ . O to B is $b$ .
4
$1 + 3 = 4 parts$ — Add the ratio numbers to find how many equal parts AB has.
5
$A M = \frac{1}{4} (b - a)$ — AM is 1 part out of 4, so it is one quarter of AB.
6
$O M = O A + A M$ — To get from O to M, go along OA, then along AM.
7
$= a + \frac{1}{4} (b - a)$ — Put in $O A = a$ and the AM we just found.
8
$= a + \frac{1}{4} b - \frac{1}{4} a$ — Multiply out the bracket.
9
$= \frac{3}{4} a + \frac{1}{4} b$ — Collect the $a$ terms: $1 - \frac{1}{4} = \frac{3}{4}$ .

Now with numbers in front of the letters, and a ratio that does not start with 1.

Worked example

1
$O A = 5 a, O B = 2 b, A X : X B = 3 : 2$ — Keep each number with its letter. X splits AB in the ratio 3 : 2.
2
$A B = - 5 a + 2 b$ — A back to O is $- 5 a$ , then O to B is $2 b$ .
3
$3 + 2 = 5 parts$ — Add the ratio numbers to find the total number of parts.
4
$A X = \frac{3}{5} (- 5 a + 2 b)$ — AX is 3 parts out of 5, so it is three fifths of AB.
5
$= - 3 a + \frac{6}{5} b$ — Multiply out: $\frac{3}{5} \times (- 5) = - 3$ and $\frac{3}{5} \times 2 = \frac{6}{5}$ .
6
$O X = O A + A X$ — To get from O to X, go along OA, then along AX.
7
$= 5 a - 3 a + \frac{6}{5} b$ — Put in $O A = 5 a$ and the AX we just found.
8
$= 2 a + \frac{6}{5} b$ — Collect the $a$ terms: $5 - 3 = 2$ .

Common mistake

Using one third for a ratio of 1 : 3. There are 4 parts in total, so AM is one quarter of AB.

Key idea

A midpoint is the ratio $1 : 1$ . Take half of the side: $A M = \frac{1}{2} (b - a)$ , so $O M = \frac{1}{2} a + \frac{1}{2} b$ .

Which Method? Read the Question

Look at the shape and what it tells you about the sides.

You see	You do
Parallelogram	opposite sides are the same vector
Trapezium, one side k times another	use k times the vector
Regular hexagon	use the long diagonal
A point in a ratio on a side	turn the ratio into a fraction
Two lines cross, find a ratio	Vector Geometry
Prove parallel or a straight line	Vector Proofs

Key idea

Write every answer in its simplest form: collect the a terms, then the b terms.

Practise expressing vectors in terms of a and b now

Interactive visual aids, then real past-paper questions marked on your working like the exam. Free to start.

Practise free on Lemmara

Real past-paper questions on expressing vectors in terms of a and b

Each with a full worked solution.

All 62 questions on this topic →

Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education or Pearson Edexcel. The lessons and worked examples are Lemmara’s own.