Editing Euclidean space (section)

==Metric structure==
The vector space <math>\overrightarrow E</math> associated to a Euclidean space {{mvar|E}} is an [[inner product space]]. This implies a [[symmetric bilinear form]]

<math display="block">\begin{align}
\overrightarrow E \times \overrightarrow E &\to \R\\
(x,y)&\mapsto \langle x,y \rangle
\end{align}</math>

that is [[positive definite]] (that is <math>\langle x,x \rangle</math> is always positive for {{math|''x'' ≠ 0}}).

The inner product of a Euclidean space is often called ''dot product'' and denoted {{math|''x'' ⋅ ''y''}}. This is specially the case when a [[Cartesian coordinate system]] has been chosen, as, in this case, the inner product of two vectors is the [[dot product]] of their [[coordinate vector]]s. For this reason, and for historical reasons, the dot notation is more commonly used than the bracket notation for the inner product of Euclidean spaces. This article will follow this usage; that is <math>\langle x,y \rangle</math> will be denoted {{math|''x'' ⋅ ''y''}} in the remainder of this article.

The '''Euclidean norm''' of a vector {{mvar|x}} is

<math display="block">\|x\| = \sqrt {x \cdot x}.</math>

The inner product and the norm allows expressing and proving [[metric space|metric]] and [[topology|topological]] properties of [[Euclidean geometry]]. The next subsection describe the most fundamental ones. ''In these subsections,'' {{mvar|E}} ''denotes an arbitrary Euclidean space, and <math>\overrightarrow E</math> denotes its vector space of translations.''

===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]].

===Orthogonality===
{{main|Perpendicular|Orthogonality}}
Two nonzero vectors {{mvar|u}} and {{mvar|v}} of <math>\overrightarrow E</math> (the associated vector space of a Euclidean space {{mvar|E}}) are ''perpendicular'' or ''orthogonal'' if their inner product is zero:

<math display="block"> u \cdot v =0</math>

Two linear subspaces of <math>\overrightarrow E</math> are orthogonal if every nonzero vector of the first one is perpendicular to every nonzero vector of the second one. This implies that the intersection of the linear subspaces is reduced to the zero vector.

Two lines, and more generally two Euclidean subspaces (A line can be considered as one Euclidean subspace.) are orthogonal if their directions (the associated vector spaces of the Euclidean subspaces) are orthogonal. Two orthogonal lines that intersect are said ''perpendicular''.

Two segments {{math|''AB''}} and {{math|''AC''}} that share a common endpoint {{math|''A''}} are ''perpendicular'' or ''form a [[right angle]]'' if the vectors <math>\overrightarrow {AB}\vphantom{\frac){}}</math> and <math>\overrightarrow {AC}\vphantom{\frac){}}</math> are orthogonal.

If {{math|''AB''}} and {{math|''AC''}} form a right angle, one has

<math display="block">|BC|^2 = |AB|^2 + |AC|^2.</math>

This is the [[Pythagorean theorem]]. Its proof is easy in this context, as, expressing this in terms of the inner product, one has, using bilinearity and symmetry of the inner product:

<math display="block">\begin{align}
|BC|^2 &= \overrightarrow {BC}\cdot \overrightarrow {BC} \vphantom{\frac({}}\\[2mu]
&=\Bigl(\overrightarrow {BA}+\overrightarrow {AC}\Bigr) \cdot \Bigl(\overrightarrow {BA}+\overrightarrow {AC}\Bigr)\\[4mu]
&=\overrightarrow {BA}\cdot \overrightarrow {BA}+ \overrightarrow {AC}\cdot \overrightarrow {AC} -2 \overrightarrow {AB}\cdot \overrightarrow {AC}\\[6mu]
&=\overrightarrow {AB}\cdot \overrightarrow {AB} + \overrightarrow {AC}\cdot\overrightarrow {AC}\\[6mu]
&=|AB|^2 + |AC|^2.
\end{align}</math>

Here, <math>\overrightarrow {AB}\cdot \overrightarrow {AC} = 0 \vphantom{\frac({}}</math> is used since these two vectors are orthogonal.

=== Angle ===
{{main article|Angle}}
[[Image:45, -315, and 405 co-terminal angles.svg|thumb|right|Positive and negative angles on the oriented plane]]
The (non-oriented) ''angle'' {{mvar|θ}} between two nonzero vectors {{mvar|x}} and {{mvar|y}} in <math>\overrightarrow E</math> is

<math display="block">\theta = \arccos\left(\frac{x\cdot y}{|x|\,|y|}\right)</math>

where {{math|arccos}} is the [[principal value]] of the [[arccosine]] function. By [[Cauchy–Schwarz inequality]], the argument of the arccosine is in the interval {{closed-closed|−1, 1}}. Therefore {{mvar|θ}} is real, and {{math|0 ≤ ''θ'' ≤ ''π''}} (or {{math|0 ≤ ''θ'' ≤ 180}} if angles are measured in degrees).

Angles are not useful in a Euclidean line, as they can be only 0 or {{pi}}.

In an [[orientation (vector space)|oriented]] Euclidean plane, one can define the ''[[oriented angle]]'' of two vectors. The oriented angle of two vectors {{mvar|x}} and {{mvar|y}} is then the opposite of the oriented angle of {{mvar|y}} and {{mvar|x}}. In this case, the angle of two vectors can have any value [[modular arithmetic|modulo]] an integer multiple of {{math|2''π''}}. In particular, a [[reflex angle]] {{math|''π'' < ''θ'' < 2''π''}} equals the negative angle {{math|−''π'' < ''θ'' − 2''π'' < 0}}.

The angle of two vectors does not change if they are [[scalar multiplication|multiplied]] by positive numbers. More precisely, if {{mvar|x}} and {{mvar|y}} are two vectors, and {{mvar|λ}} and {{mvar|μ}} are real numbers, then

<math display="block">\operatorname{angle}(\lambda x, \mu y)= \begin{cases}
\operatorname{angle}(x, y) \qquad\qquad \text{if } \lambda \text{ and } \mu \text{ have the same sign}\\
\pi - \operatorname{angle}(x, y)\qquad \text{otherwise}.
\end{cases}</math>

If {{mvar|A}}, {{mvar|B}}, and {{mvar|C}} are three points in a Euclidean space, the angle of the segments {{math|''AB''}} and {{math|''AC''}} is the angle of the vectors <math>\overrightarrow {AB}\vphantom{\frac({}}</math> and <math>\overrightarrow {AC}.\vphantom{\frac({}}</math> As the multiplication of vectors by positive numbers do not change the angle, the angle of two [[Ray (geometry)|half-line]]s with initial point {{mvar|A}} can be defined: it is the angle of the segments {{math|''AB''}} and {{math|''AC''}}, where {{mvar|B}} and {{mvar|C}} are arbitrary points, one on each half-line. Although this is less used, one can define similarly the angle of segments or half-lines that do not share an initial point.

The angle of two lines is defined as follows. If {{mvar|θ}} is the angle of two segments, one on each line, the angle of any two other segments, one on each line, is either {{mvar|θ}} or {{math|''π'' − ''θ''}}. One of these angles is in the [[interval (mathematics)|interval]] {{closed-closed|0, ''π''/2}}, and the other being in {{closed-closed|''π''/2, ''π''}}. The ''non-oriented angle'' of the two lines is the one in the interval {{closed-closed|0, ''π''/2}}. In an oriented Euclidean plane, the ''oriented angle'' of two lines belongs to the interval {{closed-closed|−''π''/2, ''π''/2}}.

===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>

=== Other coordinates ===
[[Image:Repere espace.png|thumb|right|192px|3-dimensional skew coordinates]]
{{main article|Coordinate system}}

As a Euclidean space is an [[affine space]], one can consider an [[affine frame]] on it, which is the same as a Euclidean frame, except that the basis is not required to be orthonormal. This define [[affine coordinates]], sometimes called ''skew coordinates'' for emphasizing that the basis vectors are not pairwise orthogonal.

An [[affine basis]] of a Euclidean space of dimension {{mvar|n}} is a set of {{math|''n'' + 1}} points that are not contained in a hyperplane. An affine basis define [[barycentric coordinates]] for every point.

Many other coordinates systems can be defined on a Euclidean space {{mvar|E}} of dimension {{mvar|n}}, in the following way. Let {{mvar|f}} be a [[homeomorphism]] (or, more often, a [[diffeomorphism]]) from a [[dense subset|dense]] [[open subset]] of {{mvar|E}} to an open subset of <math>\R^n.</math> The ''coordinates'' of a point {{mvar|x}} of {{mvar|E}} are the components of {{math|''f''(''x'')}}. The [[polar coordinate system]] (dimension 2) and the [[spherical coordinate system|spherical]] and [[cylindrical coordinate system|cylindrical]] coordinate systems (dimension 3) are defined this way.

For points that are outside the domain of {{mvar|f}}, coordinates may sometimes be defined as the limit of coordinates of neighbour points, but these coordinates may be not uniquely defined, and may be not continuous in the neighborhood of the point. For example, for the spherical coordinate system, the longitude is not defined at the pole, and on the [[antimeridian]], the longitude passes discontinuously from –180° to +180°.

This way of defining coordinates extends easily to other mathematical structures, and in particular to [[manifold]]s.