Editing Euclidean space (section)

===Distance and length===
{{main article|Euclidean distance}}
The ''distance'' (more precisely the ''Euclidean distance'') between two points of a Euclidean space is the norm of the translation vector that maps one point to the other; that is

<math display="block">d(P,Q) = \Bigl\|\overrightarrow {PQ}\Bigr\|.\vphantom{\frac({}}</math>

The ''length'' of a [[#Lines and segments|segment]] {{math|''PQ''}} is the distance {{math|''d''(''P'', ''Q'')}} between its endpoints ''P'' and ''Q''. It is often denoted <math>|PQ|</math>.

The distance is a [[metric (mathematics)|metric]], as it is positive definite, symmetric, and satisfies the [[triangle inequality]]

<math display="block">d(P,Q)\le d(P,R) + d(R, Q).</math>

Moreover, the equality is true if and only if a point {{mvar|R}} belongs to the segment {{math|''PQ''}}. This inequality means that the length of any edge of a [[triangle]] is smaller than the sum of the lengths of the other edges. This is the origin of the term ''triangle inequality''.

With the Euclidean distance, every Euclidean space is a [[complete metric space]].