The Math Meridian
Back to Vectors, Matrices and Conics

The dot product and the angle between vectors

Questions at easy, medium and hard levels. Pick a level and get a fresh set every time, with instant feedback.

Useful for: IB Math AA HL AP® Precalculus

Learn it u · v = u₁v₁ + u₂v₂ (+ u₃v₃ in 3D) = |u||v|cos θ. If u · v = 0 the vectors are perpendicular. Work: W = F · d.

Example: Angle between (3, 4) and (5, 0)?

  1. u · v = 15. |u| = 5, |v| = 5.
  2. cos θ = 15/25 = 0.6 → θ ≈ 53.1°.

Loading questions…