Euclidean distance

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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 ($\overline{\mathbf{p}\mathbf{q}}$).

In Cartesian coordinates, if p = (p1p2,..., pn) and q = (q1q2,..., qn) are two points in Euclidean n-space, then the distance from p to q, or from q to p is given by:

$\mathrm{d}(\mathbf{p},\mathbf{q}) = \mathrm{d}(\mathbf{q},\mathbf{p}) = \sqrt{(q_1-p_1)^2 + (q_2-p_2)^2 + \cdots + (q_n-p_n)^2} = \sqrt{\sum_{i=1}^n (q_i-p_i)^2}.$

(1)

The position of a point in a Euclidean n-space 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

$\mathbf{q} - \mathbf{p} = (q_1-p_1, q_2-p_2, \cdots, q_n-p_n)$

In a three-dimensional 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.