Maths
Vectors — addition, dot & cross products — quick study summary
A-Level MathsAP Physics CIB Maths HL
A vector has magnitude AND direction (unlike a scalar). Vectors add tip-to-tail or by adding components. The dot product a·b = |a||b|cosθ gives a scalar — useful for finding angles and for work (force·displacement). The cross product a×b = |a||b|sinθ · n̂ gives a vector perpendicular to both — useful for torque and magnetic force. Unit vectors î, ĵ, k̂ point along the x, y, z axes.
Key points
- Vectors add component-wise: (a₁,a₂) + (b₁,b₂) = (a₁+b₁, a₂+b₂)
- Magnitude: |a| = √(a₁² + a₂² + a₃²)
- Dot product: a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cosθ → scalar
- Cross product: a×b is perpendicular to both, magnitude |a||b|sinθ
- Dot product = 0 means perpendicular; cross product = 0 means parallel
Practice quiz
Click each question to reveal the answer.
1. What is the magnitude of the vector (3, 4)?
- 3
- 4
- 5
- 7
Answer: 5
|v| = √(3² + 4²) = √25 = 5.
2. What is the dot product (1, 2, 3) · (4, −5, 6)?
Answer: 12
1×4 + 2×(−5) + 3×6 = 4 − 10 + 18 = 12.
3. If a·b = 0 (both non-zero), what's the angle between a and b?
Answer: 90° (they are perpendicular)
a·b = |a||b|cosθ. Non-zero vectors with zero dot product means cosθ = 0, so θ = 90°.
Last reviewed: May 2026