dotprodukt

The dotprodukt, also known as the scalar product or inner product in broader contexts, is a mathematical operation that takes two vectors and returns a scalar. For two real vectors x = (x1, x2, ..., xn) and y = (y1, y2, ..., yn) in R^n, the dotprodukt is defined as x · y = x1y1 + x2y2 + ... + xnyn. It is denoted by x · y or ⟨x, y⟩.

Key properties include commutativity (x · y = y · x), bilinearity (x · (ay + bz) = a x · y + b x · z for scalars a, b and vectors y, z), and distributivity over addition. For a vector x, the dotprodukt with itself is nonnegative: x · x ≥ 0, with equality if and only if x = 0. The dotprodukt induces a Euclidean norm via ||x|| = sqrt(x · x). The relation between dotprodukt and angle θ between x and y is x · y = ||x|| ||y|| cos θ.

In complex vector spaces, the dotprodukt generalizes to ⟨x, y⟩ = Σ xi conj(yi), which is conjugate-symmetric and used to define norms and angles there as well.

Applications are widespread. The dotprodukt is used to compute projections, test for orthogonality, and measure similarity between vectors (e.g., cosine similarity). It underpins many algorithms in computer graphics, physics, engineering, and machine learning (as a linear kernel). Computationally, it requires O(n) time for vectors of length n and can be implemented efficiently on hardware.

Generalizations include inner products on abstract vector spaces, which satisfy similar properties and define geometry beyond Euclidean space.

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