In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space (or even any inner product space) becomes a metric space. The associated norm is called the Euclidean norm. Older literature refers to the metric as Pythagorean metric.
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Definition
The Euclidean distance between points p and q is the length of the line segment connecting them ().
In Cartesian coordinates, if p = (p_{1}, p_{2},..., p_{n}) and q = (q_{1}, q_{2},..., q_{n}) are two points in Euclidean nspace, then the distance from p to q, or from q to p is given by:
(1)
The position of a point in a Euclidean nspace is an Euclidean vector. So, p and q are Euclidean vectors, starting from the origin of the space, and their tips indicate two points. The Euclidean norm, or Euclidean length, or magnitude of a vector measures the length of the vector:
where the last equation involves the dot product.
A vector can be described as a directed line segment from the origin of the Euclidean space (vector tail), to a point in that space (vector tip). If we consider that its length is actually the distance from its tail to its tip, it becomes clear that the Euclidean norm of a vector is just a special case of Euclidean distance: the Euclidean distance between its tail and its tip.
The distance between points p and q may have a direction (e.g. from p to q), so it may be represented by another vector, given by
In a threedimensional space (n=3), this is an arrow from p to q, which can be also regarded as the position of q relative to p. It may be also called a displacement vector if p and q represent two positions of the same point at two successive instants of time.
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