Editing Euclidean space (section)

===Cartesian coordinates===
{{See also|Cartesian coordinate system}}

Every Euclidean vector space has an [[orthonormal basis]] (in fact, infinitely many in dimension higher than one, and two in dimension one), that is a [[basis (vector space)|basis]] <math>(e_1, \dots, e_n) </math> of [[unit vector]]s (<math>\|e_i\| = 1</math>) that are pairwise orthogonal (<math>e_i\cdot e_j = 0</math> for {{math|''i'' ≠ ''j''}}). More precisely, given any [[basis (vector space)|basis]] <math>(b_1, \dots, b_n),</math> the [[Gram–Schmidt process]] computes an orthonormal basis such that, for every {{mvar|i}}, the [[linear span]]s of <math>(e_1, \dots, e_i)</math> and <math>(b_1, \dots, b_i)</math> are equal.<ref>{{harvtxt|Anton|1987|pp=209–215}}</ref>

Given a Euclidean space {{mvar|E}}, a ''[[Cartesian frame]]'' is a set of data consisting of an orthonormal basis of <math>\overrightarrow E,</math> and a point of {{mvar|E}}, called the ''origin'' and often denoted {{mvar|O}}. A Cartesian frame <math>(O, e_1, \dots, e_n)</math> allows defining Cartesian coordinates for both {{mvar|E}} and <math>\overrightarrow E</math> in the following way.

The Cartesian coordinates of a vector {{mvar|v}} of <math>\overrightarrow E</math> are the coefficients of {{mvar|v}} on the orthonormal basis <math>e_1, \dots, e_n.</math> For example, the Cartesian coordinates of a vector <math>v</math> on an orthonormal basis <math>(e_1,e_2,e_3)</math> (that may be named as <math>(x,y,z)</math> as a convention) in a 3-dimensional Euclidean space is <math>(\alpha_1,\alpha_2,\alpha_3)</math> if <math>v = \alpha_1 e_1 + \alpha_2 e_2 + \alpha_3 e_3</math>. As the basis is orthonormal, the {{mvar|i}}-th coefficient <math>\alpha_i</math> is equal to the dot product <math>v\cdot e_i.</math>

The Cartesian coordinates of a point {{mvar|P}} of {{mvar|E}} are the Cartesian coordinates of the vector <math>\overrightarrow {OP}.\vphantom{\frac({}}</math>