Editing Euclidean vector (section)

===Euclidean and affine vectors===
In the geometrical and physical settings, it is sometimes possible to associate, in a natural way, a ''length'' or magnitude and a direction to vectors. In addition, the notion of direction is strictly associated with the notion of an ''angle'' between two vectors. If the [[dot product]] of two vectors is defined—a scalar-valued product of two vectors—then it is also possible to define a length; the dot product gives a convenient algebraic characterization of both angle (a function of the dot product between any two non-zero vectors) and length (the square root of the dot product of a vector by itself). In three dimensions, it is further possible to define the [[cross product]], which supplies an algebraic characterization of the [[area]] and [[orientation (geometry)|orientation]] in space of the [[parallelogram]] defined by two vectors (used as sides of the parallelogram). In any dimension (and, in particular, higher dimensions), it is possible to define the [[exterior product]], which (among other things) supplies an algebraic characterization of the area and orientation in space of the ''n''-dimensional [[parallelepiped#Parallelotope|parallelotope]] defined by ''n'' vectors.

In a [[pseudo-Euclidean space]], a vector's squared length can be positive, negative, or zero. An important example is [[Minkowski space]] (which is important to our understanding of [[special relativity]]).

However, it is not always possible or desirable to define the length of a vector. This more general type of spatial vector is the subject of [[vector space]]s (for free vectors) and [[affine space]]s (for bound vectors, as each represented by an ordered pair of "points"). One physical example comes from [[thermodynamics]], where many quantities of interest can be considered vectors in a space with no notion of length or angle.<ref name="thermo-forms"
>[http://www.av8n.com/physics/thermo-forms.htm Thermodynamics and Differential Forms]</ref>