G10GeometryHigher

Vectors

A vector has both size and direction. At GCSE you write them as column vectors and add, subtract and scale them — then use them to prove things about shapes (a Higher-tier favourite).

40 min AQA GCSE Maths

What you'll learn

Write and read column vectors
Add and subtract vectors
Multiply a vector by a scalar
Use vectors in simple geometry proofs

Column vectors

A column vector $(3 - 2)$ means 3 right and 2 down. Add or subtract vectors by working on the top and bottom numbers separately.

(a b) + (c d) = (a + c b + d)

Add components separately.

1Top:

3 + 2 = 5

2Bottom:

1 + 4 = 5

Answer

(55)

Scaling and notation

Multiplying a vector by a number scales both components: $2 (31) = (62)$ . Vectors are written in $bold$ (or underlined by hand), and $A B$ means the vector from A to B.

Tip — Parallel vectors are scalar multiples of each other — useful in proofs.

Remember these

(a b) + (c d) = (a + c b + d)

Adding vectors.

k (a b) = (k a k b)

Scalar multiple.

Watch out for these

Adding the top of one vector to the bottom of another.

Add tops to tops and bottoms to bottoms.

Forgetting a negative means the opposite direction.

−AB is the same length as AB but points the other way.

Key takeaways

Column vector (a, b) = a across, b up/down.
Add/subtract components separately; scale by multiplying both.
Parallel vectors are scalar multiples.

Test yourself

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