Vector (mathematics and physics)

In mathematics and physics, a vector is an element of a vector space.
For many specific vector spaces, the vectors have received specific names, which are listed below.
Historically, vectors were introduced in geometry and physics (typically in mechanics) before the formalization of the concept of vector space. Therefore, one often talks about vectors without specifying the vector space to which they belong. Specifically, in a Euclidean space, one considers "spatial vectors", also called "Euclidean vectors" which are used to represent quantities that have both magnitude and direction, and may be added, subtracted and scaled (i.e. multiplied by a real number) for forming a vector space.
Vectors in Euclidean geometry.
In classical Euclidean geometry (i.e., synthetic geometry), vectors were introduced (during the 19th century) as equivalence classes under equipollence, of ordered pairs of points; two pairs and being equipollent if the points , in this order, form a parallelogram. Such an equivalence class is called a "vector", more precisely, a Euclidean vector. The equivalence class of is often denoted formula_1
A Euclidean vector is thus an equivalence class of directed segments with the same magnitude (e.g., the length of the line segment ) and same direction (e.g., the direction from to ). 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 scalars, which have no direction. For example, velocity, forces and acceleration are represented by vectors.
In modern geometry, Euclidean spaces are often defined from linear algebra. More precisely, a Euclidean space is defined as a set to which is associated an inner product space of finite dimension over the reals formula_2 and a group action of the additive group of formula_2 which is free and transitive (See Affine space for details of this construction). The elements of formula_4 are called 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 formula_5 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, formula_5 is a Euclidean space, with itself as an associated vector space, and the dot product as an inner product.
The Euclidean space formula_5 is often presented as "the" Euclidean space of dimension . This is motivated by the fact that every Euclidean space of dimension is isomorphic to the Euclidean space formula_8 More precisely, given such a Euclidean space, one may choose any point as an 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 of the space, as the coordinates on this basis of the vector formula_9 These choices define an isomorphism of the given Euclidean space onto formula_10 by mapping any point to the -tuple of its Cartesian coordinates, and every vector to its coordinate vector.
Tuples that are not really vectors.
The set formula_5 of tuples of real numbers has a natural structure of vector space defined by component-wise addition and scalar multiplication. When such tuples are used for representing some data, it is common to call them "vectors", even if the vector addition does not mean anything for these data, which may make the terminology confusing. Similarly, some physical phenomena involve a direction and a magnitude. They are often represented by vectors, even if operations of vector spaces do not apply to them.
Vectors in algebras.
Every algebra over a field is a vector space, but elements of an algebra are generally not called vectors. However, in some cases, they are called "vectors", mainly due to historical reasons.
See also.
Vector fields.
A vector field is a vector-valued function that, generally, has a domain of the same dimension (as a manifold) as its codomain,