Vectors in 3D, lines and planes

In three dimensions, vectors describe lines and planes in space. The dot product measures angles, and the cross product gives a vector perpendicular to two others.

Key facts

  • Line: r = a + λd (a point plus multiples of a direction).
  • Dot product: a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ.
  • Perpendicular vectors have dot product 0.
  • Plane: r · n = d, with n normal to the plane.

Worked examples

Example 1

Find the angle between (1, 2, 2) and (2, −1, 2).

  1. a · b = 2 − 2 + 4 = 4
  2. |a| = 3, |b| = 3
  3. cos θ = 4/9 θ ≈ 63.6°

Answer: 63.6°