Dot product

The dot product is a scalar formed by multiplying corresponding vector components and summing the products. In Euclidean space, it encodes vector lengths and the cosine of the angle between them.

Euclidean inner product: A symmetric, positive-definite bilinear form that assigns a scalar to each pair of vectors in a real vector space. The dot product is the standard inner product on real coordinate space.
Vector: An element of a vector space, often represented geometrically by magnitude and direction. The dot product takes two vectors as its inputs.
Cosine similarity: A measure of directional similarity between vectors, calculated as their dot product divided by their lengths. It normalizes the dot product to compare orientation independently of vector magnitude.
Cross product: A three-dimensional vector product whose result is perpendicular to both input vectors and whose magnitude depends on their angle. Unlike the scalar dot product, it returns a vector encoding oriented area.
Cauchy–Schwarz inequality: An inequality bounding the absolute inner product of two vectors by the product of their lengths. It constrains dot products and guarantees the cosine formula stays between −1 and 1.
Scalar multiplication: The operation of multiplying a vector by a scalar to produce another vector. Projection and vector-length formulas use scalar multiples of vectors.
Least squares: A method for fitting a model by minimizing the sum of squared residuals. Normal equations use inner products to characterize the best-fitting solution.
Hadamard product: An elementwise product of vectors or matrices that multiplies corresponding entries without summing them. It uses the same componentwise multiplication but leaves the products as separate entries.
Vector projection: The component of one vector along another, represented by a scaled copy of the second vector. Dividing a dot product by a vector’s squared length gives the projection coefficient.
Vector addition: The operation combining two vectors by adding their corresponding components. Bilinearity means dot products distribute over vector sums.

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