Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Vectors

In one sentence. Column vectors, magnitude, and the a-and-b algebra of geometric proof — the topic that ends most Higher papers. Higher Tier.

Vectors & Transformations — one of 2 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Higher Tier only.

This module ends with Simon Stevin. Learn Vectors, teach it, then defend it against Simon Stevin.

Where this sits in the specification

Specification 4MA1, 5.1 Vectors (Higher). Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 5.1 — Vectors (Higher)

1 · Understand it

What it is

A vector has length and direction: as a column (3, −2) means 3 right, 2 down. Vectors add tip-to-tail, scale by numbers (2a is twice as long, −a points back), and |v| = √(x² + y²) gives magnitude. In geometry, journeys are vector sums: AB means 'from A to B', and any route between the same endpoints gives the same vector.

Why it works

Routes compose: going A→O then O→C is the same displacement as A→C, so AC = AO + OC — algebra you can walk along. Parallelism becomes multiplication: if one vector is a scalar multiple of another (PQ = 2a + 4b = 2(a + 2b)), they're parallel; sharing a point besides makes them collinear. That's how vector PROOF works.

When a question is asking for it

Column-vector arithmetic and magnitudes; then the classic endgame question: a figure defined by a and b, asked to express some segment and prove two segments parallel or three points collinear. Route-finding — rewriting a journey via known legs — is the whole technique.

How to hold it in your head

Read AB as the instruction 'travel from A to B', and re-route freely through known points: A→B = A→O→B = −a + b. Any legal route works because only start and finish matter. To reverse a leg, flip its sign.

2 · The mistakes that cost marks

  • AB read as A's position or as 'B to A'. It's the journey FROM A TO B — direction is part of the object.
  • AB computed as a − b. Via the origin: AB = −OA + OB = b − a. (The far end minus the near end.)
  • Adding magnitudes of non-parallel vectors: |a + b| ≠ |a| + |b| unless they point the same way.
  • Claiming parallel without exhibiting the scalar: the proof IS writing one vector as k × the other.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

p = (−2, 3) and q = (5, 1). Work out 2p − 3q.

  1. Scale first: 2p = (−4, 6) and 3q = (15, 3).
  2. Subtract component by component: (−4 − 15, 6 − 3) = (−19, 3).

Answer: (−19, 3)

OACB is a parallelogram with OA = a and OB = b. M is the midpoint of AC. Express OM in terms of a and b.

  1. Route O → A → M, through points whose legs are known.
  2. AC is parallel to OB and the same length (opposite sides of a parallelogram), so AC = b.
  3. M is halfway along AC, so AM = ½b.
  4. OM = OA + AM = a + ½b.

Answer: OM = a + ½b

OA = a and OB = b. D is the point with OD = 3b − 2a. Prove that A, B and D lie on a straight line.

  1. AB = −a + b = b − a (far end minus near end).
  2. AD = −a + (3b − 2a) = 3b − 3a.
  3. Factorise to expose the multiple: AD = 3(b − a) = 3AB.
  4. So AD is a multiple of AB: the lines are parallel. They also share the point A, so they are the same line.

Answer: AD = 3AB, so AD is parallel to AB and they share point A: A, B and D are collinear.

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — a = (3, −2) and b = (−1, 4), written as column vectors. Work out a + 2b. (Type it like (1,6).)
  2. Grade 7 — Work out the magnitude of the vector (6, −8).
  3. Grade 8 — OABC is a parallelogram with OA = a and OC = c. Express the vector AC in terms of a and c. (Type it like c-a.)
  4. Grade 9 — OAB is a triangle with OA = a and OB = b. M is the midpoint of AB. Express OM in terms of a and b. (Type it like (a+b)/2.)
  5. 9+ — OA = a and OB = b. Point P lies on AB with AP : PB = 2 : 1. Express OP in terms of a and b. (Type it like (a+2b)/3.)

Ask the person who made it

Inside Incandio you can put a question to James Clerk Maxwell, who answers knowing exactly which module you have just finished: “Your equations describe fields with both size AND direction — quantities that ordinary numbers can't hold. Why do electricity, magnetism and force need vectors, and what does a vector let you say that a number can't?”

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Vectors, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Vectors works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Vectors against Simon Stevin — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 9 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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Carry on through the course