Editing Euclidean vector (section)

===Further information===

In classical [[Euclidean geometry]] (i.e., [[synthetic geometry]]), vectors were introduced (during the 19th century) as [[equivalence class]]es under [[Equipollence (geometry)|equipollence]], of [[ordered pair]]s of points; two pairs {{math|(''A'', ''B'')}} and {{math|(''C'', ''D'')}} being equipollent if the points {{math|''A'', ''B'', ''D'', ''C''}}, in this order, form a [[parallelogram]]. Such an equivalence class is called a ''vector'', more precisely, a Euclidean vector.<ref>In some old texts, the pair {{math|(''A'', ''B'')}} is called a ''bound vector'', and its equivalence class is called a ''free vector''.</ref> The equivalence class of {{math|(''A'', ''B'')}} is often denoted <math>\overrightarrow{AB}.</math>

A Euclidean vector is thus an equivalence class of directed segments with the same magnitude (e.g., the length of the [[line segment]] {{math|(''A'', ''B'')}}) and same direction (e.g., the direction from {{mvar|A}} to {{mvar|B}}).<ref name="1.1: Vectors">{{Cite web|date=2013-11-07|title=1.1: Vectors|url=https://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Vector_Calculus/1%3A_Vector_Basics/1.1%3A_Vectors|access-date=2020-08-19|website=Mathematics LibreTexts|language=en}}</ref> In physics, Euclidean vectors are used to represent physical quantities that have both magnitude and direction, but are not located at a specific place, in contrast to [[Scalar (mathematics)|scalar]]s, which have no direction.<ref name=":2"/> For example, [[velocity]], [[force]]s and [[acceleration]] are represented by vectors.

In modern geometry, Euclidean spaces are often defined from [[linear algebra]]. More precisely, a Euclidean space {{mvar|E}} is defined as a set to which is associated an [[inner product space]] of finite dimension over the reals <math>\overrightarrow{E},</math> and a [[Group action (mathematics)|group action]] of the [[additive group]] of <math>\overrightarrow{E},</math> which is [[free action|free]] and [[transitive action|transitive]] (See [[Affine space]] for details of this construction). The elements of <math>\overrightarrow{E}</math> are called [[translation (geometry)|translations]]. It has been proven that the two definitions of Euclidean spaces are equivalent, and that the equivalence classes under equipollence may be identified with translations.

Sometimes, Euclidean vectors are considered without reference to a Euclidean space. In this case, a Euclidean vector is an element of a normed vector space of finite dimension over the reals, or, typically, an element of the [[real coordinate space]] <math>\mathbb R^n</math> equipped with the [[dot product]]. This makes sense, as the addition in such a vector space acts freely and transitively on the vector space itself. That is, <math>\mathbb R^n</math> is a Euclidean space, with itself as an associated vector space, and the dot product as an inner product.

The Euclidean space <math>\mathbb R^n</math> is often presented as ''the'' [[standard Euclidean space]] of dimension {{mvar|n}}. This is motivated by the fact that every Euclidean space of dimension {{mvar|n}} is [[isomorphism|isomorphic]] to the Euclidean space <math>\mathbb R^n.</math> More precisely, given such a Euclidean space, one may choose any point {{mvar|O}} as an [[origin (geometry)|origin]]. By [[Gram–Schmidt process]], one may also find an [[orthonormal basis]] of the associated vector space (a basis such that the inner product of two basis vectors is 0 if they are different and 1 if they are equal). This defines [[Cartesian coordinates]] of any point {{mvar|P}} of the space, as the coordinates on this basis of the vector <math>\overrightarrow{OP}.</math> These choices define an isomorphism of the given Euclidean space onto <math>\mathbb R^n,</math> by mapping any point to the [[tuple|{{mvar|n}}-tuple]] of its Cartesian coordinates, and every vector to its [[coordinate vector]].