📐 Angle

The angle between vectors and can be calculated by

Intuitively, the smaller the angle, the more similar and are.

Orthogonality §

Then, since orthogonal vectors have , two vectors are orthogonal if and only if . If and are unit, they’re orthonormal.

Furthermore, a basis is orthonormal if for and . It’s possible to construct an orthonormal basis from any basis using the Gram-Schmidt process.

The orthogonal complement of subspace contains all vectors in that are orthogonal to every vector in . The intersection of and is the vector, and the bases of each combined form a basis for .