A-Level Maths: Vectors Revision

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What is the magnitude of the vector (3, 4)?

Vectors is a core part of A-Level Maths. Revise the key concepts and common mistakes below, then lock them in with the free games.

Key concepts

VectorQuantity with magnitude and direction. Written as column or i-j form: a = (3, 4) or 3i + 4j.

Magnitude|a| = √(x² + y²) for a = xi + yj. Distance in Pythagorean form.

Unit vectorVector of length 1 in direction of a: â = a/|a|. Used to specify direction without magnitude.

Vector additionAdd component-wise: (a, b) + (c, d) = (a + c, b + d). Geometrically: tip-to-tail or parallelogram rule.

Scalar multiplek·(x, y) = (kx, ky). Parallel vectors are scalar multiples of each other.

Position vectorsPosition of point P relative to origin O written as OP. AB = OB − OA gives displacement from A to B.

Dot producta·b = a₁b₁ + a₂b₂ + a₃b₃ — a scalar (number).

Geometric forma·b = |a||b|cosθ where θ is the angle between vectors.

Perpendicular testIf a·b = 0 and neither is zero, the vectors are perpendicular.

Parallel vectorsa parallel to b means a = kb for some scalar k.

Angle findingcos θ = (a·b)/(|a||b|), then θ = arccos.

Self dota·a = |a|², so magnitude squared.

Vector line equationr = a + λb where a is a point on the line and b is the direction.

Direction vectorAny non-zero scalar multiple of b also works as direction.

What your exam board covers

Where Vectors sits in each exam board's specification. Tap a board to see what it lists.

AQAVectorsSpec 7357

Part of: Pure mathematics

Source: AQA A-level Mathematics (7357) specification — Subject content, accessed 23 June 2026.

CCEAVectorsSpec 2210

Part of: AS 1 — Pure mathematics

Source: CCEA A-Level Mathematics specification, accessed 23 June 2026.

EdexcelVectorsSpec 9MA0

Part of: Pure mathematics

Source: Pearson Edexcel A level Mathematics (9MA0) specification, accessed 23 June 2026.

OCRVectorsSpec H240

Part of: Pure mathematics

Source: OCR A Level Mathematics A (H240) specification, accessed 23 June 2026.

Learn Vectors step by step

Free interactive mini-lessons: short screens, worked examples and quick checks. Pick your exam board:

Common mistakes to avoid

Questions where students often pick the tempting wrong answer — make sure you know the right one:

What is the magnitude of the vector (3, 4)?✗ The magnitude is 7 — you add the components. ✓ 5 — using sqrt(3^2 + 4^2).

If the dot product of two non-zero vectors is zero, what does that mean?✗ It means at least one of the vectors is the zero vector. ✓ The vectors are perpendicular to each other.

Practise Vectors — free games

Test yourself with these quick revision games for this topic:

▶ Two Truths, One Lie ▶ Reveal Race ▶ Connections Grid ▶ Falling Words ▶ Word Web

See all 22 games in the Revision Arcade →

Vectors

Key concepts

What your exam board covers

Common mistakes to avoid

Practise Vectors — free games

More A-Level Maths topics