Orthogonal matrices and the dot product

An orthogonal matrix — a square matrix A with Aᵀ A = 1 — preserves the dot product of vectors, and in particular their squared lengths. This is the algebraic content behind the frame-independence of the rotational kinetic energy in rigid-body dynamics: rotating a velocity does not change its speed.

An orthogonal matrix preserves the dot product: if Aᵀ A = 1 then (A v) ⬝ᵥ (A w) = v ⬝ᵥ w. Taking w = v shows orthogonal matrices preserve squared lengths, (A v) ⬝ᵥ (A v) = v ⬝ᵥ v.