Lagevektor

Lagevektor, often called the position vector, is the vector that describes the location of a point relative to a chosen origin in a Euclidean space. For a point P with Cartesian coordinates (x1, x2, ..., xn) in space with origin O, the Lagevektor r points from O to P.

In 2D, r = (x, y); in 3D, r = (x, y, z). It can be written in vector form as r = x1 e1 + x2 e2 + ... + xn en, where {e1, ..., en} is a chosen basis, typically the standard basis.

Properties: r is an element of R^n, its magnitude |r| is the distance from the origin to P, and its direction is from the origin toward P. The displacement between two points P and Q is r_Q − r_P. Position vectors can be added and scaled, and the dot product with another vector yields projections and angles.

Coordinate systems: The components of the Lagevektor depend on the chosen basis; when switching bases, components transform accordingly using the basis vectors. In physics, the time derivative of the Lagevektor gives velocity, and the second derivative gives acceleration.

Example: In 2D, a point with coordinates (3, 4) has Lagevektor r = (3, 4) and magnitude |r| = 5. The distance from the origin to P is 5 units.

See also: Ortsvektor, position vector, displacement, Cartesian coordinate system, vector.

Etymology: The term Lagevektor is used in Dutch and some German-language mathematical texts to denote the location of a point relative to an origin.