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Concept
Vector (mathematics and physics)
Summary
In mathematics and physics, vector is a term that refers colloquially to some quantities that cannot be expressed by a single number (a scalar), or to elements of some vector spaces.
Historically, vectors were introduced in geometry and physics (typically in mechanics) for quantities that have both a magnitude and a direction, such as displacements, forces and velocity. Such quantities are represented by geometric vectors in the same way as distances, masses and time are represented by real numbers.
The term vector is also used, in some contexts, for tuples, which are finite sequences of numbers of a fixed length.
Both geometric vectors and tuples can be added and scaled, and these vector operations led to the concept of a vector space, which is a set equipped with a vector addition and a scalar multiplication that satisfy some axioms generalizing the main properties of operations on the above sorts of vectors. A vector space formed by geometric vectors is called a Euclidean vector space, and a vector space formed by tuples is called a coordinate vector space.
Many vector spaces are considered in mathematics, such as extension field, polynomial rings, algebras and function spaces. The term vector is generally not used for elements of these vectors spaces, and is generally reserved for geometric vectors, tuples, and elements of unspecified vector spaces (for example, when discussing general properties of vector spaces).
Euclidean vector
Vector space
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.
Vector quaternion, a quaternion with a zero real part
Multivector or p-vector, an element of the exterior algebra of a vector space.
Spinors, also called spin vectors, have been introduced for extending the notion of rotation vector. In fact, rotation vectors represent well rotations locally, but not globally, because a closed loop in the space of rotation vectors may induce a curve in the space of rotations that is not a loop.
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