Editing Euclidean space (section)

===Parallelism===
{{main|Parallel (geometry)}}

Two subspaces {{mvar|S}} and {{mvar|T}} of the same dimension in a Euclidean space are ''parallel'' if they have the same direction (i.e., the same associated vector space).{{efn|It may depend on the context or the author whether a subspace is parallel to itself}} Equivalently, they are parallel, if there is a translation vector {{mvar|v}} that maps one to the other:

<math display="block">T= S+v.</math>

Given a point {{mvar|P}} and a subspace {{mvar|S}}, there exists exactly one subspace that contains {{mvar|P}} and is parallel to {{mvar|S}}, which is <math>P + \overrightarrow S.</math> In the case where {{mvar|S}} is a line (subspace of dimension one), this property is [[Playfair's axiom]].

It follows that in a Euclidean plane, two lines either meet in one point or are parallel.

The concept of parallel subspaces has been extended to subspaces of different dimensions: two subspaces are parallel if the direction of one of them is contained in the direction to the other.

{{anchor|Euclidean norm}}