Euclidean space

A real affine space equipped with a positive-definite inner product on its displacement vectors. This structure defines distance, angle, and perpendicularity independently of any choice of origin or coordinates.

Affine space: A set of points with a vector space of displacements, where differences are defined but no origin is distinguished. It supplies points and displacement vectors before a metric is added.
Pythagorean theorem: A relation stating that the square of a right triangle's hypotenuse equals the sum of the squares of its legs. Orthogonal displacement components combine by this rule to give squared distance.
Minkowski space: A real vector space with a nondegenerate indefinite bilinear form, used to model spacetime in special relativity. Its metric allows nonzero vectors of zero length, unlike Euclidean space.
Euclidean geometry: The study of geometric properties preserved by Euclidean transformations, including lengths, angles, and incidence. It develops the geometric consequences of the space's metric and affine structure.
Real vector space: A set with addition and scalar multiplication by real numbers, obeying the vector space axioms. Displacement vectors in Euclidean space form a real vector space.
Dot product: A bilinear product of vectors whose value encodes their lengths and mutual angle in Euclidean coordinates. It computes lengths and angles from displacement vectors.
Pseudo-Euclidean space: A real affine space whose displacement vectors carry a nondegenerate indefinite quadratic form. It replaces positive-definite lengths with a metric of mixed signs.
Cartesian coordinate system: A coordinate system that assigns points numerical coordinates relative to chosen axes and an origin. Orthonormal Cartesian coordinates express Euclidean distances with the Pythagorean formula.
Inner product space: A real or complex vector space equipped with an inner product that defines lengths and angles. A positive-definite inner product on displacements provides the Euclidean metric.
Orthogonality: The relation between vectors whose inner product is zero. It formalizes perpendicularity for displacement vectors.

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