GCSE Maths Vectors Notes
A vector has both size and direction. Column vectors add and subtract component-by-component, and parallel vectors are multiples of each other.
Key concepts
- Column vector
- Written as a bracket with an x-move on top and a y-move below; positive is right/up.
- Adding vectors
- Add the top numbers and the bottom numbers separately (nose-to-tail on a diagram).
- Scalar multiple
- Multiplying a vector by a number scales its length; the result is parallel to the original.
- Parallel vectors
- Two vectors are parallel if one is a multiple of the other — a key idea in vector proofs.
Key formulas
Magnitude
|(a, b)| = √(a² + b²)
Length of a column vector, from Pythagoras.
Worked example
Given a = (3, 1) and b = (−1, 4), find 2a − b as a column vector.
- 1Double a: 2a = (6, 2).
- 2Subtract b component-by-component: (6 − (−1), 2 − 4).
- 3Simplify: (7, −2).
Answer: 2a − b = (7, −2)
Common mistakes
✗ Adding the top of one vector to the bottom of another.
✓ Keep x with x and y with y — top with top, bottom with bottom.
✗ Ignoring the direction of a vector, e.g. AB vs BA.
✓ BA is the negative of AB — reverse the sign of both components.
✗ Not stating 'parallel' in a proof.
✓ Show one vector is a scalar multiple of the other, then conclude they are parallel.
Quick check
Try these, then reveal the answers.
(2, 5) + (3, −1) = ?Reveal answer
(5, 4)
Magnitude of (3, 4)?Reveal answer
5
Is (4, 6) parallel to (2, 3)?Reveal answer
Yes — it is 2 × (2, 3).
Exam tips
- Draw the vectors nose-to-tail to check your route around a shape is correct.
- In proofs, write each path as a sum of given vectors, then simplify and compare.
- A shared common factor between two vectors is your evidence that they are parallel.
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Frequently asked questions
What does the magnitude of a vector mean?+
It is the length of the vector, found with Pythagoras from its horizontal and vertical components.
How do I prove three points are collinear?+
Show two vectors along the line are parallel and share a common point, so they lie on one straight line.
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