DotProduct

The dot product, also called the scalar product, is a mathematical operation that takes two vectors of the same dimension and returns a scalar. For a = (a1, ..., an) and b = (b1, ..., bn), the dot product is a · b = a1b1 + a2b2 + ... + anbn. In Euclidean space R^n with the standard coordinates, this is the familiar sum of pairwise products. In real vector spaces the dot product is commutative (a · b = b · a) and bilinear (linear in each argument).

Key properties include a · a ≥ 0 with equality only when a = 0, which leads to the norm ||a|| = sqrt(a · a). Orthogonality is defined by a · b = 0. The dot product also satisfies the Cauchy–Schwarz inequality, |a · b| ≤ ||a|| ||b||.

Geometrically, a · b = ||a|| ||b|| cos θ, where θ is the angle between a and b. It also represents the length of the projection of a onto the direction of b, namely (a · b)/||b||. When b is a unit vector, a · b is the component of a in the direction of b.

Computationally, in coordinates a simple 2D example is a = (1, 2) and b = (3, 4); a · b = 1×3 + 2×4 = 11. The dot product can be written as a^T b in matrix form. In complex spaces, the standard inner product uses conjugation: a · b = Σ ai conj(bi).

Related concepts include inner products in abstract vector spaces, Gram–Schmidt orthogonalization, and cosine similarity in data analysis, which normalizes the dot product by the norms of the vectors. Applications span physics, computer graphics, and machine learning.