Vectors — AQA GCSE Maths Revision Notes
What you'll learn
This topic covers vectors: quantities that carry both a size and a direction. By the end of this guide you should be able to read and write vector notation, add and subtract vectors, multiply a vector by a number, and find a vector's magnitude.
You should also be able to describe a journey through a shape in terms of given vectors, show that two lines are parallel, and show that three points lie on a straight line.
The organising idea is that a vector is a journey, not a place. The vector AB means "the trip from A to B" — it records how far and in which direction you travelled, but says nothing about where you started. Everything in the topic follows from that. Going backwards is the negative of going forwards. Two journeys done one after another add up. A journey twice as long in the same direction is twice the vector. And a route through an intermediate point can always replace a direct one, which is the whole technique behind vector proofs.
Key terms and definitions
Vector — a quantity with both magnitude and direction.
Scalar — an ordinary number, with size but no direction.
Column vector — a vector written as two numbers in a bracket, one above the other: the top is the movement right, the bottom is the movement up.
Magnitude — the length of a vector, written with vertical bars around it.
Resultant — the single vector equivalent to two or more added together.
Parallel vectors — vectors that are multiples of one another.
Collinear points — three or more points lying on the same straight line.
Core concepts
Notation
A vector can be written in three ways, and exam papers use all of them.
As a bold or underlined letter, such as a. When writing by hand, underline it, since you cannot write in bold.
As two capital letters with an arrow above, such as AB, meaning the journey from A to B.
As a column vector, written vertically in a bracket. A column vector of 3 over 2 means 3 to the right and 2 up. Negative entries reverse those directions: −3 over 2 means 3 to the left and 2 up.
Note that a column vector is not a coordinate, even though both use brackets. A coordinate is a place; a vector is a movement.
Reversing a vector
Travelling from B back to A undoes the journey from A to B, so BA = −AB.
In column form, reversing changes the sign of both entries: the reverse of 3 over 2 is −3 over −2.
This single fact does most of the work in vector proofs, because it lets any journey be run backwards when needed.
Adding vectors
Two journeys done one after another combine into a single journey called the resultant.
In column form, add the tops and add the bottoms separately. Adding 3 over 1 to 2 over −4 gives 5 over −3.
Geometrically the vectors are placed nose to tail: the second starts where the first finishes, and the resultant runs from the very start to the very end.
The rule that makes proofs work is that you may always travel via another point: AB = AC + CB. Going from A to C and then C to B gets you to the same place as going straight from A to B.
Subtracting vectors
Subtracting is adding the reverse, so AB − CD means AB + (−CD).
In column form, subtract the tops and subtract the bottoms.
A useful result follows from combining the two rules above. Since AB = AO + OB and AO = −OA, we get AB = OB − OA: the vector between two points is the far one minus the near one.
Multiplying by a scalar
Multiplying a vector by a number multiplies both entries, which changes the length but not the line it lies along.
So 3a is three times as long as a, in the same direction. And −2a is twice as long in the opposite direction.
A scalar between 0 and 1 shortens the vector: ½a is half as long, still in the same direction.
Magnitude
The magnitude is the length of the vector, found using Pythagoras' theorem, because the two entries of a column vector are the two shorter sides of a right-angled triangle.
For a column vector of 3 over 4, the magnitude is the square root of 3² + 4², which is the square root of 25, so 5.
Note that magnitude is always positive, and that a vector and its negative have the same magnitude — reversing a journey does not change how far you travelled.
Proving lines are parallel
Two vectors are parallel if one is a scalar multiple of the other. Nothing more is required.
So if one journey works out as 2a + 4b and another as a + 2b, the first is exactly twice the second, and the two lines are parallel.
Write the conclusion out in full: "AB = 2 CD, so AB is parallel to CD." The statement of the multiple is what earns the mark, not the algebra alone.
Proving points are collinear
Three points are collinear if two of the vectors joining them are parallel and share a common point.
Showing AB = 3 BC proves that A, B and C lie on a straight line, because the two journeys run along the same direction and B is on both.
The shared point matters. Two parallel vectors with no point in common describe two separate parallel lines, not one straight line.
Worked examples
Example 1: Column vector arithmetic
Let a be the column vector 4 over 1 and b be the column vector −2 over 3. Find a + 2b, and give its magnitude to one decimal place.
First find 2b by doubling both entries: −4 over 6.
Add a to it, taking tops and bottoms separately: 4 + (−4) = 0, and 1 + 6 = 7. So a + 2b is the column vector 0 over 7.
The magnitude is the square root of 0² + 7², which is 7.0. The horizontal movement cancelled completely, leaving a purely vertical journey.
Example 2: A journey through a shape
In a shape, OA = a and OB = b. Express AB in terms of a and b.
You cannot travel from A to B directly using the vectors given, so go via O.
AB = AO + OB.
Now AO is the reverse of OA, so AO = −a.
Therefore AB = −a + b, which is usually written b − a.
This matches the general result that the vector between two points is the far one minus the near one.
Example 3: Proving lines are parallel
In a figure, PQ = 3a + 6b and RS = a + 2b. Show that PQ is parallel to RS.
Look for a common factor in PQ: 3a + 6b = 3(a + 2b).
The bracket is exactly RS, so PQ = 3 RS.
Since PQ is a scalar multiple of RS, PQ is parallel to RS, and it is three times as long.
The conclusion needs both parts written out: the multiple, and the word parallel.
Common mistakes and how to avoid them
Treating a column vector as a coordinate. A coordinate is a position; a vector is a movement. The brackets look alike but mean different things.
Getting the direction of a reversed vector wrong. BA = −AB. Reversing changes the sign of both entries.
Multiplying only one entry by a scalar. Both parts are multiplied.
Giving a negative magnitude. Magnitude is a length, so it is never negative, and a vector and its negative have the same magnitude.
Stopping a proof at the algebra. State that one vector is a scalar multiple of the other, and therefore that the lines are parallel.
Confusing parallel with collinear. Collinear also requires a shared point.
Losing track of direction in a journey. Write each leg as a separate labelled step rather than doing it in your head.
Exam technique for "Vectors"
Write every journey as a sum of labelled legs before simplifying — "AB = AO + OB" on its own line. Method marks are awarded for a correct route even when the simplification afterwards is wrong.
Underline vectors when writing by hand, so the examiner can see which quantities are vectors and which are scalars.
Look for a common factor whenever a proof asks about parallel lines. That factor is the answer.
State conclusions in words. "PQ = 3 RS, so PQ is parallel to RS and three times its length" earns marks that the algebra alone does not.
Take the route through a point you have a vector for, even when it looks longer. A detour through the origin is almost always available.
Check a magnitude for sense: it must be positive, and at least as large as either entry on its own.
Quick revision summary
A vector is a journey, not a place: AB is the trip from A to B, and it says nothing about where you started.
BA = −AB. Reversing a journey negates both entries of its column vector.
To add column vectors, add tops and bottoms separately; geometrically the vectors go nose to tail and the total is the resultant. You may always travel via another point: AB = AC + CB, which gives the useful result AB = OB − OA.
Multiplying by a scalar multiplies both entries, changing the length but not the line: 3a is three times as long, −2a twice as long the other way.
The magnitude comes from Pythagoras — for 3 over 4 it is 5 — and is always positive.
Two vectors are parallel when one is a scalar multiple of the other. Three points are collinear when two such vectors also share a common point.
In a proof, look for a common factor, then state the multiple and the conclusion in words.