IGCSE Vectors Explained

What is a vector?

A vector describes both direction and magnitude. Moving six squares right and four up is different from moving six left and four down, even though the distance travelled is equal. Cambridge IGCSE Mathematics 0580 includes translation by a column vector in Core (C7.1). Extended students also study vector arithmetic (E7.2), magnitude (E7.3) and vector geometry (E7.4).

1. Read a column vector

A column vector has two entries, one above the other. Here (6, 4)ᵀ represents the column vector with 6 on top and 4 underneath. The first number describes horizontal movement and the second describes vertical movement. (6, 4)ᵀ means six right and four up; (−3, 2)ᵀ means three left and two up. A displacement describes movement, not the destination: the starting point matters.

2. Find AB from two points

Let A = (−2, 1) and B = (4, 5). Subtract the coordinates of A from those of B:

AB = (4 − (−2), 5 − 1)ᵀ = (6, 4)ᵀ.

Check: start at A, move six right and four up, and arrive at B. The reverse vector BA = (−6, −4)ᵀ has equal length but opposite direction. In general, BA = −AB. A frequent error is subtracting in the wrong order.

3. Translate a point

Suppose P = (3, −1) is translated by (−5, 2)ᵀ. Add the components to the coordinates:

P′ = (3 − 5, −1 + 2) = (−2, 1).

For a triangle, apply the same movement to all three vertices. A translation preserves shape and size.

4. Add, subtract and scale vectors

Let a = (3, −2)ᵀ and b = (−1, 4)ᵀ.

Add corresponding components: a + b = (2, 2)ᵀ.

Scale both components: 2a = (6, −4)ᵀ.

Subtract: 2a − b = (6 − (−1), −4 − 4)ᵀ = (7, −8)ᵀ.

These operations combine movements. The path rule AB + BC = AC says that travelling from A to B and then B to C produces the same displacement as travelling directly from A to C.

5. Magnitude is length

The components of a vector form a right-angled triangle. For v = (6, 4)ᵀ:

|v| = √(6² + 4²) = √52 = 2√13 ≈ 7.21 units.

The vector gives direction and length; its magnitude is a nonnegative number. The reverse vector has the same magnitude. A common mistake is reporting 52 instead of √52: 52 is the squared length.

6. Position vectors and midpoint

A position vector locates a point relative to the origin O. If A = (2, 1) and B = (8, 5), then OA = (2, 1)ᵀ and OB = (8, 5)ᵀ. The midpoint M is:

M = ((2 + 8)/2, (1 + 5)/2) = (5, 3).

In vector notation, OM = (OA + OB)/2. Check that AM = (3, 2)ᵀ and MB = (3, 2)ᵀ.

7. Prove three points are collinear

Points are collinear if they lie on one straight line. Take A = (1, 2), B = (4, 6), C = (10, 14).

AB = (3, 4)ᵀ and AC = (9, 12)ᵀ.

Since AC = 3AB, both vectors from A point in the same direction, so A, B and C are collinear. Also BC = (6, 8)ᵀ = 2AB, giving AB : BC = 1 : 2.

A complete proof should state the scalar-multiple relationship and why it proves collinearity.

Three original practice questions

Q1: A = (−1, 3), B = (5, −5). Find AB and its magnitude.

Answer: AB = (6, −8)ᵀ and magnitude √(36 + 64) = 10.

Q2: Translate P = (2, −6) by (−3, 4)ᵀ.

Answer: (−1, −2).

Q3: a = (2, −1)ᵀ, b = (−4, 3)ᵀ. Find 3a + b.

Answer: (6, −3)ᵀ + (−4, 3)ᵀ = (2, 0)ᵀ.

Study tips

Sketch each displacement on a grid and check its direction before calculating.

Try translations, coordinate differences, vector arithmetic and magnitudes before midpoint and collinearity proofs. If an arrow points in the wrong direction, revisit the concept before moving to harder algebra.

Official source: Cambridge IGCSE Mathematics 0580 syllabus 2025–2027, sections C7.1 and E7.2–E7.4.

Additional source: Cambridge vectors teaching pack.

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