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.
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.
What does the gradient of a straight-line graph tell you?✗ Where the line crosses the y-axis. ✓ How steep the line is — the change in y for each unit change in x (rise over run).
If you divide both sides of an inequality by -2, what happens to the inequality sign?✗ The inequality sign stays the same — you treat it like an equation. ✓ The inequality sign reverses (e.g. > becomes <).
Practise Vectors — free games
Test yourself with these quick revision games for this topic:
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