GCSE Maths vectors

By The STEM-Tutor team · Updated October 2026

Vectors appear on GCSE Maths Higher papers, often as a final multi-mark question. Here is how to approach them.

What a vector is

A vector has a size and a direction. In GCSE Maths it is written as a column vector, as a bold letter such as a, or as an arrow between two points such as AB with an arrow over it.

Column vectors

• The top number is the movement right (negative means left).
• The bottom number is the movement up (negative means down).
• Add vectors by adding the tops and the bottoms.
• Multiply by a number by multiplying both parts.

Vector paths

To find a vector between two points, follow a route along vectors you know. Going backwards along a vector reverses its sign: BA = −AB.

Parallel vectors

Two vectors are parallel if one is a multiple of the other. For example, 2a + 4b is parallel to a + 2b.

Vector proof questions

1. Write each vector you need in terms of a and b, step by step.
2. Simplify carefully.
3. To show three points lie on a straight line, show two vectors between them are parallel and share a point.
4. Finish with a clear sentence that answers the question.

Common mistakes

• Forgetting to reverse the sign when going backwards.
• Mixing up ratios: if M is the midpoint of AB, AM is ½AB.
• Showing vectors are parallel but not stating they share a point.

Thinking of vectors as journeys

A helpful way to picture a vector is as a journey: how far to go and in which direction. The column vector (3, 2) means three steps right and two steps up. Adding vectors means doing one journey after another, and the result is the overall journey from start to finish. Going backwards along a vector changes its sign, so the journey from B to A is the negative of the journey from A to B.

Worked examples

Example 1: adding column vectors

a = (3, −1) and b = (−2, 5). Find a + b and 2a − b.

• a + b = (3 + −2, −1 + 5) = (1, 4).
• 2a − b = (6, −2) − (−2, 5) = (8, −7).

Example 2: magnitude

Find the length of the vector (5, 12).

• Use Pythagoras: √(25 + 144) = √169 = 13.

Example 3: a vector path

OA = a and OB = b. Find AB.

• Go from A back to O, then O to B: AB = −a + b = b − a.

Example 4: a midpoint

M is the midpoint of AB, with OA = a and OB = b. Find OM.

• OM = OA + AM = a + ½(b − a) = ½a + ½b.

Example 5: a ratio

P lies on AB so that AP : PB = 2 : 1. Find OP.

• AP is 2/3 of AB, so AP = ⅔(b − a).
• OP = a + ⅔(b − a) = ⅓a + ⅔b.

Example 6: proving points lie on a line

AB = 2a + b and AC = 6a + 3b. Show that A, B and C lie on a straight line.

• AC = 3(2a + b) = 3AB, so AC is parallel to AB.
• They also share the point A, so A, B and C are collinear. You must state both facts for full marks.

How vector proof questions are marked

Vector proof questions are usually worth several marks, and most of those marks are for method. Write each vector you find as a separate line, using the route you took. When you show two vectors are parallel, write one as a multiple of the other, such as “XY = 2PQ, so XY is parallel to PQ”. When you show points are collinear, add the shared point. Clear, step-by-step working is the difference between a partial answer and full marks.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Are vectors on the Foundation paper?

Column vectors, adding and multiplying by a number can appear on Foundation. Vector proofs are Higher tier.

How do I write vectors by hand?

Underline the letter, such as a, because you cannot write in bold. For a vector between points, draw an arrow over the letters, such as AB with an arrow.

Key takeaways

• A vector has size and direction.
• Add column vectors by adding the top and bottom numbers.
• AB = b − a.
• Parallel vectors are multiples of each other.
• For collinear points, show parallel and a shared point.

Where to go next

Vectors link to Pythagoras and to further study at A level.

• GCSE Maths trigonometry
• GCSE Maths topics
• A level Maths vectors

More mistakes to avoid

• Getting the sign wrong when going against the direction of a vector.
• Forgetting to simplify, such as leaving ½(b − a) + a instead of ½a + ½b.
• Stating that vectors are parallel without writing one as a multiple of the other.
• Mixing up ratios: if AP : PB = 2 : 1, then AP is 2/3 of AB, not 1/2.

A full vector proof, step by step

Here is how a typical Higher tier proof fits together. OAB is a triangle with OA = a and OB = b. M is the midpoint of OA and N is the midpoint of OB. Show that MN is parallel to AB.

First, find AB: AB = b − a. Next, find MN by going from M to O and then from O to N: MN = −½a + ½b = ½(b − a). Finally, compare: MN = ½AB, so MN is parallel to AB and half its length. This is the midpoint theorem, proved in four lines with vectors.

Vectors can feel abstract at first, but with a little practice they become one of the most predictable topics on the paper.

Example 7: a vector in a hexagon

ABCDEF is a regular hexagon with centre O. OA = a and OB = b. Find AB and OC.

• AB = b − a.
• In a regular hexagon, OC is parallel and equal to AB, so OC = b − a. Spotting equal vectors in shapes like this saves a lot of work.

Regular shapes such as hexagons and parallelograms contain many equal and parallel vectors, so look for them before calculating a long route.