Vector (mathematics and physics)

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.

The term vector is also used, in some contexts, for tuples, which are finite sequences (of numbers or other objects) of a fixed length.

Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors,[6] operations which obey the familiar algebraic laws of commutativity, associativity, and distributivity.

For example, a position vector in physical space may be expressed as three Cartesian coordinates with SI unit of meters.

For example, an event in spacetime may be represented as a position four-vector, with coherent derived unit of meters: it includes a position Euclidean vector and a timelike component, t ⋅ c0 (involving the speed of light).

A vector may also result from the evaluation, at a particular instant, of a continuous vector-valued function (e.g., the pendulum equation).

of tuples of n real numbers has a natural structure of vector space defined by component-wise addition and scalar multiplication.

It is common to call these tuples vectors, even in contexts where vector-space operations do not apply.

Vector-valued functions, where the output is a vector, are scrutinized using calculus to derive essential insights into motion within three-dimensional space.

Line integrals, crucial for calculating work along a path within force fields, and surface integrals, employed to determine quantities like flux, illustrate the practical utility of calculus in vector analysis.

Volume integrals, essential for computations involving scalar or vector fields over three-dimensional regions, contribute to understanding mass distribution, charge density, and fluid flow rates.

[citation needed] A vector field is a vector-valued function that, generally, has a domain of the same dimension (as a manifold) as its codomain,