# 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 ${\mathbb{R}}^{n}$ as vector space).

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

If $a=({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})$ and $b=({y}_{1},{y}_{2},\mathrm{\dots},{y}_{n})$, then formula 1 can be generalized to ${\mathbb{R}}^{n}$ by defining the Euclidean distance from $a$ to $b$ as

$$d(a,b)=\sqrt{{({x}_{1}-{y}_{1})}^{2}+{({x}_{2}-{y}_{2})}^{2}+\mathrm{\cdots}+{({x}_{n}-{y}_{n})}^{2}}.$$ | (2) |

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

This can also be done for ${\u2102}^{n}$ since as set $\u2102={\mathbb{R}}^{2}$ and thus the metric on $\u2102$ is the same given to ${\mathbb{R}}^{2}$, and in general, ${\u2102}^{n}$ 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 |