Vectors

GCSE Maths · Geometry

Vectors (Higher)

A vector describes a movement with both magnitude (size) and direction. Unlike a number (scalar), a vector tells you which way to go.

Notation

Vectors can be written as:

A bold lowercase letter: a (in handwriting, underline: a̲)
A column vector: (x, y) — x units right and y units up
Between points: the vector from A to B is written →AB

Column Vectors

The vector (3, −2) means move 3 right and 2 down.

Adding vectors: Add the components separately. (2, 5) + (3, −1) = (5, 4)

Subtracting vectors: Subtract the components. (4, 7) − (1, 3) = (3, 4)

Scalar multiplication: Multiply each component. 3 × (2, −1) = (6, −3)

Vector Arithmetic with Letters

If a and b are vectors:

a + b: travel along a then along b
a − b: travel along a then backwards along b (equivalent to a + (−b))
2a: travel along a twice (same direction, double the length)
−a: travel in the opposite direction to a

Finding Vectors in Diagrams

The vector →AB means "how do I get from A to B?" You can build a path using known vectors.

Key rule: →AB = −→BA (reversing direction negates the vector)

Worked Example: In triangle OAB, →OA = a and →OB = b. M is the midpoint of AB. Find →OM.

→AB = →AO + →OB = −a + b = b − a
→AM = ½→AB = ½(b − a)
→OM = →OA + →AM = a + ½(b − a) = a + ½b − ½a = ½a + ½b

Or equivalently: →OM = ½(a + b)

Parallel Vectors

Two vectors are parallel if one is a scalar multiple of the other.

If →PQ = k × →RS for some scalar k, then PQ is parallel to RS.

Worked Example: Show that the line joining the midpoints of two sides of a triangle is parallel to the third side.

Let →OA = a, →OB = b. M is midpoint of OA, N is midpoint of OB.

→MN = →MO + →ON = −½a + ½b = ½(b − a) = ½→AB
Since →MN = ½→AB, MN is parallel to AB and half its length. ∎

Proving Points are Collinear

Three points are collinear (on the same straight line) if the vector from one to another is a scalar multiple of the vector between another pair, AND they share a common point.

Worked Example: →OA = 2a + 3b, →OB = 5a + 6b, →OC = 11a + 12b. Show A, B, C are collinear.

→AB = →OB − →OA = 3a + 3b = 3(a + b)
→AC = →OC − →OA = 9a + 9b = 9(a + b)
→AC = 3 × →AB

Since →AC is a scalar multiple of →AB and they share point A, A, B and C are collinear. ∎

Magnitude of a Vector

The magnitude (length) of vector (x, y) is: |v| = √(x² + y²)

Worked Example: Find the magnitude of (5, −12).

|v| = √(25 + 144) = √169 = 13

Exam Tips

Always show your route through the diagram clearly — e.g. "→OM = →OA + →AM"
To prove parallel, show one vector is a scalar multiple of the other — state this conclusion explicitly
To prove collinear, you need parallel vectors AND a shared point
Diagrams are not always to scale — rely on the algebra, not the picture