Euclidean vector

A Euclidean vector is an element of a real vector space equipped with an inner product. Its length and angles are defined by that product, independently of any coordinate system.

Vector space: A set with vector addition and scalar multiplication satisfying the vector space axioms. Euclidean vectors belong to a vector space, with an inner product adding geometric structure.
Vector addition: The operation that combines two vectors into their sum. It combines displacements and obeys the parallelogram rule in Euclidean geometry.
Displacement (geometry): A vector describing the change from one position to another. Displacements add head-to-tail and model Euclidean vectors directly.
Affine space: A space of points in which differences are vectors but points themselves have no distinguished origin. A Euclidean vector is based at the origin; an affine point is not itself a vector.
Inner product: A bilinear, symmetric, positive-definite function that assigns a real number to each pair of vectors. It defines the lengths and angles that make a vector space Euclidean.
Scalar multiplication: The operation that multiplies a vector by a real number, scaling its magnitude and possibly reversing its direction. It changes a vector's length while keeping it on the same line.
Force: A physical interaction that can change an object's motion and is represented by magnitude and direction. In classical mechanics, forces combine according to vector addition.
Complex vector space: A vector space whose scalars are complex numbers. Its scalar field differs, and its inner product convention requires care with complex conjugation.
Euclidean norm: The length of a vector induced by an inner product, equal to the square root of its inner product with itself. It gives a Euclidean vector its magnitude.
Dot product: The coordinate formula for the standard inner product of real vectors: the sum of products of corresponding components. It calculates lengths, angles, and projections from coordinates.

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