# Euclidean distance

If $u=(x_{1},y_{1})$ and $v=(x_{2},y_{2})$ are two points on the plane, their Euclidean distance is given by

 $\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}.$ (1)

Geometrically, it’s the length of the segment joining $u$ and $v$, and also the norm of the difference vector (considering $\mathbbmss{R}^{n}$ as vector space).

This distance  induces a metric (and therefore a topology  ) on $\mathbbmss{R}^{2}$, called Euclidean metric (on $\mathbbmss{R}^{2}$) or standard metric (on $\mathbbmss{R}^{2})$. The topology so induced is called standard topology or usual topology on $\mathbbmss{R}^{2}$ and one basis can be obtained considering the set of all the open balls.

If $a=(x_{1},x_{2},\ldots,x_{n})$ and $b=(y_{1},y_{2},\ldots,y_{n})$, then formula 1 can be generalized to $\mathbbmss{R}^{n}$ by defining the Euclidean distance from $a$ to $b$ as

 $d(a,b)=\sqrt{(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}+\cdots+(x_{n}-y_{n})^{2}}.$ (2)

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

This can also be done for $\mathbbmss{C}^{n}$ since as set $\mathbbmss{C}=\mathbbmss{R}^{2}$ and thus the metric on $\mathbbmss{C}$ is the same given to $\mathbbmss{R}^{2}$, and in general, $\mathbbmss{C}^{n}$ gets the same metric as $R^{2n}$.

 Title Euclidean distance Canonical name EuclideanDistance Date of creation 2013-03-22 12:08:21 Last modified on 2013-03-22 12:08:21 Owner drini (3) Last modified by drini (3) Numerical id 15 Author drini (3) Entry type Definition Classification msc 53A99 Classification msc 54E35 Synonym Euclidean metric Synonym standard metric Synonym standard topology Synonym Euclidean Synonym canonical topology Synonym usual topology Related topic Topology Related topic BoundedInterval Related topic EuclideanVectorSpace Related topic DistanceOfNonParallelLines Related topic EuclideanVectorSpace2 Related topic Hyperbola2 Related topic CassiniOval