Editing Euclidean vector (section)

==Overview==
In [[physics]] and [[engineering]], a vector is typically regarded as a geometric entity characterized by a [[magnitude (mathematics)|magnitude]] and a [[relative direction]]. It is formally defined as a [[directed line segment]], or arrow, in a [[Euclidean space]].<ref>{{harvnb|Itô|1993|p=1678}}</ref> In [[pure mathematics]], a [[vector (mathematics)|vector]] is defined more generally as any element of a [[vector space]]. In this context, vectors are abstract entities which may or may not be characterized by a magnitude and a direction. This generalized definition implies that the above-mentioned geometric entities are a special kind of abstract vectors, as they are elements of a special kind of vector space called [[Euclidean space]]. This particular article is about vectors strictly defined as arrows in Euclidean space. When it becomes necessary to distinguish these special vectors from vectors as defined in pure mathematics, they are sometimes referred to as '''''geometric''''', '''''spatial''''', or '''''Euclidean''''' vectors.

A Euclidean vector may possess a definite ''initial point'' and ''terminal point''; such a condition may be emphasized calling the result a '''''bound vector'''''.<ref>Formerly known as ''located vector''. See {{harvnb|Lang|1986|page=9}}.</ref> When only the magnitude and direction of the vector matter, and the particular initial or terminal points are of no importance, the vector is called a '''''free vector'''''. The distinction between bound and free vectors is especially relevant in mechanics, where a [[force]] applied to a body has a point of contact (see [[resultant force]] and [[Couple (mechanics)|couple]]).

Two arrows <math>\stackrel {\,\longrightarrow}{AB}</math> and <math>\stackrel {\,\longrightarrow}{A'B'}</math> in space represent the same free vector if they have the same magnitude and direction: that is, they are [[equipollence (geometry)|equipollent]] if the quadrilateral ''ABB′A′'' is a [[parallelogram]]. If the Euclidean space is equipped with a choice of [[origin (mathematics)|origin]], then a free vector is equivalent to the bound vector of the same magnitude and direction whose initial point is the origin.

The term ''vector'' also has generalizations to higher dimensions, and to more formal approaches with much wider applications.

===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]].

===Examples in one dimension===
Since the physicist's concept of [[force (physics)|force]] has a direction and a magnitude, it may be seen as a vector. As an example, consider a rightward force ''F'' of 15 [[Newton (unit)|newtons]]. If the positive [[Cartesian coordinate system|axis]] is also directed rightward, then ''F'' is represented by the vector 15&nbsp;N, and if positive points leftward, then the vector for ''F'' is &minus;15&nbsp;N. In either case, the magnitude of the vector is 15&nbsp;N. Likewise, the vector representation of a displacement Δ''s'' of 4 [[meter (unit)|meters]] would be 4&nbsp;m or &minus;4&nbsp;m, depending on its direction, and its magnitude would be 4&nbsp;m regardless.

===In physics and engineering===
Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has magnitude, has direction, and which adheres to the rules of vector addition. An example is [[velocity]], the magnitude of which is [[speed]]. For instance, the velocity ''5 meters per second upward'' could be represented by the vector (0, 5) (in 2 dimensions with the positive ''y''-axis as 'up'). Another quantity represented by a vector is [[force]], since it has a magnitude and direction and follows the rules of vector addition.<ref name=":2" /> Vectors also describe many other physical quantities, such as linear displacement, [[displacement (vector)|displacement]], linear acceleration, [[angular acceleration]], [[linear momentum]], and [[angular momentum]]. Other physical vectors, such as the [[electric field|electric]] and [[magnetic field]], are represented as a system of vectors at each point of a physical space; that is, a [[vector field]]. Examples of quantities that have magnitude and direction, but fail to follow the rules of vector addition, are angular displacement and electric current. Consequently, these are not vectors.

===In Cartesian space===
In the [[Cartesian coordinate system]], a bound vector can be represented by identifying the coordinates of its initial and terminal point. For instance, the points {{math|''A'' {{=}} (1, 0, 0)}} and {{math|''B'' {{=}} (0, 1, 0)}} in space determine the bound vector <math>\overrightarrow{AB}</math> pointing from the point {{math|''x'' {{=}} 1}} on the ''x''-axis to the point {{math|''y'' {{=}} 1}} on the ''y''-axis.

In Cartesian coordinates, a free vector may be thought of in terms of a corresponding bound vector, in this sense, whose initial point has the coordinates of the origin {{math|''O'' {{=}} (0, 0, 0)}}. It is then determined by the coordinates of that bound vector's terminal point. Thus the free vector represented by (1, 0, 0) is a vector of unit length—pointing along the direction of the positive ''x''-axis.

This coordinate representation of free vectors allows their algebraic features to be expressed in a convenient numerical fashion. For example, the sum of the two (free) vectors (1, 2, 3) and (−2, 0, 4) is the (free) vector
<math display=block>(1, 2, 3) + (-2, 0, 4) = (1-2, 2+0, 3+4) = (-1, 2, 7)\,.</math>

===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>

===Generalizations===
In physics, as well as mathematics, a vector is often identified with a [[tuple]] of components, or list of numbers, that act as scalar coefficients for a set of [[basis vector]]s. When the basis is transformed, for example by rotation or stretching, then the components of any vector in terms of that basis also transform in an opposite sense. The vector itself has not changed, but the basis has, so the components of the vector must change to compensate. The vector is called ''covariant'' or ''contravariant'', depending on how the transformation of the vector's components is related to the transformation of the basis. In general, contravariant vectors are "regular vectors" with units of distance (such as a displacement), or distance times some other unit (such as velocity or acceleration); covariant vectors, on the other hand, have units of one-over-distance such as [[gradient]]. If you change units (a special case of a [[change of basis]]) from meters to millimeters, a scale factor of 1/1000, a displacement of 1&nbsp;m becomes 1000&nbsp;mm—a contravariant change in numerical value. In contrast, a gradient of 1&nbsp;[[Kelvin|K]]/m becomes 0.001&nbsp;K/mm—a covariant change in value (for more, see [[covariance and contravariance of vectors]]). [[Tensor]]s are another type of quantity that behave in this way; a vector is one type of [[tensor]].

In pure [[mathematics]], a vector is any element of a [[vector space]] over some [[field (mathematics)|field]] and is often represented as a [[coordinate vector]]. The vectors described in this article are a very special case of this general definition, because they are contravariant with respect to the ambient space. Contravariance captures the physical intuition behind the idea that a vector has "magnitude and direction".