Euclidean distance (original) (raw)

If u=(x1,y1) and v=(x2,y2) are two points on the plane, their Euclidean distance is given by

Geometrically, it’s the length of the segment joining u and v, and also the norm of the difference vector (considering ℝn as vector space).

This distance induces a metric (and therefore a topology) on ℝ2, called Euclidean metric (on R2) or standard metric (on R2). The topology so induced is called standard topology or usual topology on R2 and one basis can be obtained considering the set of all the open balls.

If a=(x1,x2,…,xn) and b=(y1,y2,…,yn), then formula 1 can be generalized to ℝn by defining the Euclidean distance from a to b as

Notice that this distance coincides with absolute value when n=1. Euclidean distance on ℝn is also a metric (Euclidean or standard metric), and therefore we can give ℝn a topology, which is called the standard (canonical, usual, etc) topology of ℝn. The resulting (topological and vectorial) space is known as Euclidean space.

This can also be done for ℂn since as set ℂ=ℝ2 and thus the metric on ℂ is the same given to ℝ2, and in general, ℂn gets the same metric as R2⁢n.