Dot product: angles

The dot product (or scalar product) of two vectors gives a number:

a · b = |a| |b| cos θ, where θ is the angle between them (0 ≤ θ ≤ π).

In components it is quick: multiply matching components and add. a · b = a₁b₁ + a₂b₂ + a₃b₃.

Putting the two together gives the angle between two vectors:

cos θ = (a · b) / (|a| |b|)

Example: a = î + ĵ and b = ĵ + k̂. a · b = 0 + 1 + 0 = 1. |a| = |b| = √2. cos θ = 1/2, so θ = 60° (π/3).

Perpendicular test: cos 90° = 0, so two non-zero vectors are perpendicular exactly when a · b = 0. Example: (2î + ĵ – k̂) · (î – ĵ + k̂) = 2 – 1 – 1 = 0, so they are perpendicular.

The sign of a · b tells you about the angle: positive means acute (0 to 90°), zero means right angle, negative means obtuse.

Projection of a on b: how much of a lies along b, like a shadow. It equals (a · b)/|b|. Example: the projection of 3î + 4ĵ on î is 3, the î part.

For the unit vectors: î · î = ĵ · ĵ = k̂ · k̂ = 1, and î · ĵ = ĵ · k̂ = k̂ · î = 0.