GCSE Maths • Foundation & Higher
GCSE Vectors Revision Guide
Work with vector components, journeys and algebraic notation, then use vectors to prove parallel lines, collinear points and geometric relationships.
Foundation
Exam practice
Worked examples
What You'll Learn
Understand Vector Notation
Recognise vectors as quantities with both magnitude and direction, and read vectors written in column form or represented by letters such as a and b.
Add and Subtract Vectors
Add or subtract corresponding horizontal and vertical components accurately, including values involving negative numbers.
Multiply Vectors by Scalars
Scale a vector by multiplying every component by the same ordinary number.
Follow Vector Journeys
Combine movements such as AB and BC to find AC, and reverse a vector correctly by changing its sign.
Use Algebraic Vector Expressions
Work confidently with expressions such as a + b, 2a and a − b.
Prove Geometric Relationships
Use scalar multiples and vector reasoning to show that lines are parallel, points are collinear and midpoint relationships are correct.
Key Rules and Formulas
Vector Lessons
Vector Arithmetic
Learn how to add, subtract and scale column vectors, use letter notation, and combine vectors to describe journeys and positions.
Geometric Vector Proof
Use vector arithmetic and algebraic reasoning to prove geometric facts, including parallel lines, collinear points, midpoint relationships and properties of shapes.
CORE SUBTOPICS
Writing and Interpreting Vectors
Read column vectors and letter vectors, and interpret their horizontal/vertical movement, magnitude and direction.
Adding Column Vectors
Add corresponding components to combine movements.
Subtracting Column Vectors
Subtract corresponding components and handle negative numbers accurately.
Scalar Multiplication
Multiply every component by the same scalar to change the vector’s size and, for negative scalars, its direction.
Vector Proof in Shapes
Use known side vectors in shapes such as parallelograms to find diagonals and prove relationships.
Parallel Lines
Show that one vector is a scalar multiple of another.
Worked Example
Finding the Resultant Vector
QUESTION
Given AB = (2, 3) and BC = (4, 1), find AC.
- Given information: The vector AB shows the movement from A to B, and BC shows the movement from B to C.
- Method: To find AC, add the two vectors component by component.
- Add the horizontal components: 2 + 4 = 6.
- Add the vertical components: 3 + 1 = 4.
- So, AC = (6, 4).
- This gives the overall movement from A to C.
- Final Answer: AC = (6, 4).
- Key idea: When vectors form a journey, the total vector is found by adding the connected vectors. So: AC = AB + BC.
- Common mistake: do not subtract the vectors here. Since the journey goes from A to B and then B to C, the vectors must be added, not subtracted.
- Common GCSE Mistakes
- Mixing up vector components.
- Sign errors with negative numbers.
- Forgetting to multiply every component by a scalar.
- Ignoring vector direction.
- Not showing why vectors are parallel or collinear.
- Exam Tips
- Check vector direction carefully.
- Show component calculations clearly.
- For position vectors, use AB = OB − OA.
- Show scalar multiples when proving parallel or collinear vectors.
- Finish proof questions with a clear conclusion.
Ready to Practice?
Printable Worksheet
Practise simplifying, unit conversions, sharing totals, recipe scaling, combined ratios and best-buy comparisons.
Interactive Quiz
Check ratio order, equivalent ratios, total parts, unit prices and common misconceptions.
Exam-Style Questions
Apply ratio methods to money, measures, recipes, maps and multi-stage GCSE contexts.
Watch the Vectors Tutorial
Clear, step-by-step explanation of expanding brackets, simplifying complex expressions, and solving for x. Perfect for visual learners who need a breakdown of the rules in action.
- 12 minute comprehensive guide
- High-definition whiteboard walk-through
Continue Your GCSE Maths Revision
Continue Your GCSE Maths Revision
Frequently Asked Questions
A vector is a quantity with both magnitude and direction. In GCSE Maths, vectors often describe movement and can be written in column form or represented by letters such as a and b.
Work component by component. Add or subtract the horizontal/top components together and then the vertical/bottom components together.
A scalar is an ordinary number. Multiplying a vector by a scalar means multiplying every component of the vector by that number.
Use AB = OB − OA. The order is important because reversing the direction changes the sign of the vector.
