# Vector quantity

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Physical quantity that is a vector

In the [natural sciences](/source/Natural_sciences), a **vector quantity** (also known as a **vector physical quantity**, **physical vector**, or simply **vector**) is a [vector](/source/Vector_(mathematics_and_physics))-valued [physical quantity](/source/Physical_quantity).[1][2] It is typically formulated as the product of a *[unit of measurement](/source/Unit_of_measurement)* and a *vector [numerical value](/source/Numerical_value)* ([unitless](/source/Unitless)), often a [Euclidean vector](/source/Euclidean_vector) with [magnitude](/source/Vector_norm) and [direction](/source/Direction_(geometry)). For example, a [position vector](/source/Position_vector) in [physical space](/source/Physical_space) may be expressed as [three](/source/Three_dimensional) [Cartesian coordinates](/source/Cartesian_coordinates) with [SI unit](/source/SI_unit) of [meters](/source/Meters).

In [physics](/source/Physics) and [engineering](/source/Engineering), particularly in [mechanics](/source/Mechanics), a physical vector may be endowed with additional structure compared to a geometrical vector.[3] A **bound vector** is defined as the combination of an ordinary vector quantity and a *[point of application](/source/Point_of_application)* or *point of action*.[1] [4] Bound vector quantities are formulated as a *[directed line segment](/source/Directed_line_segment)*, with a definite initial point besides the magnitude and direction of the main vector.[1][3] For example, a [force](/source/Force) on the [Euclidean plane](/source/Euclidean_plane) has two Cartesian components in SI unit of [newtons](/source/Newtons) (describing the magnitude and direction of the force) and an accompanying two-dimensional position vector in meters (describing the point of application of the force), for a total of four numbers on the plane (and six in space).[5][6][4] A simpler example of a bound vector is the [translation](/source/Translation_(geometry)) vector from an initial point to an end point; in this case, the bound vector is an [ordered pair](/source/Ordered_pair) of points in the same position space, with all coordinates having the same [quantity dimension](/source/Quantity_dimension) and unit (length and meters).[7][8] A **sliding vector** is the combination of an ordinary vector quantity and a *[line of application](/source/Line_of_application)* or *line of action*, over which the vector quantity can be translated (without rotations). A **free vector** is a vector quantity having an undefined point or region of application; it can be freely translated with no consequences; a [displacement vector](/source/Displacement_vector) is a prototypical example of free vector.

Aside from the notion of units and support, physical vector quantities may also differ from Euclidean vectors in terms of [metric](/source/Distance_metric). For example, an event in [spacetime](/source/Spacetime) may be represented as a [position four-vector](/source/Position_four-vector), with [coherent derived unit](/source/Coherent_derived_unit) of meters: it includes a position Euclidean vector and a [timelike](/source/Timelike) component, *t*⋅*c*0 (involving the [speed of light](/source/Speed_of_light)). In that case, the [Minkowski metric](/source/Minkowski_metric) is adopted instead of the [Euclidean metric](/source/Euclidean_metric).

Vector quantities are a generalization of [scalar quantities](/source/Scalar_quantity) and can be further generalized as [tensor quantities](/source/Tensor_quantity).[8] Individual vectors may be ordered in a [sequence](/source/Sequence) over time (a [time series](/source/Time_series)), such as position vectors [discretizing](/source/Discretizing) a [trajectory](/source/Trajectory). A vector may also result from the [evaluation](/source/Function_evaluation), at a particular instant, of a continuous [vector-valued function](/source/Vector-valued_function) (e.g., the [pendulum equation](/source/Pendulum_equation)). In the natural sciences, the term "vector quantity" also encompasses *[vector fields](/source/Vector_field)* defined over a [two-](/source/Two-dimensional) or three-dimensional [region](/source/Region_(mathematics)) of space, such as [wind velocity](/source/Wind_velocity) over Earth's surface. [Pseudo vectors](/source/Pseudo_vector) and [bivectors](/source/Bivector) are also admitted as physical vector quantities.

## See also

- [List of vector quantities](/source/List_of_vector_quantities)

- [Vector representation](/source/Vector_representation)

## References

1. ^ [***a***](#cite_ref-a306_1-0) [***b***](#cite_ref-a306_1-1) [***c***](#cite_ref-a306_1-2) ["Details for IEV number 102-03-21: "vector quantity""](https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=102-03-21). *International Electrotechnical Vocabulary* (in Japanese). Retrieved 2024-09-07.

1. **[^](#cite_ref-o531_2-0)** ["Details for IEV number 102-03-04: "vector""](https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=102-03-04). *International Electrotechnical Vocabulary* (in Japanese). Retrieved 2024-09-07.

1. ^ [***a***](#cite_ref-m813_3-0) [***b***](#cite_ref-m813_3-1) Rao, A. (2006). [*Dynamics of Particles and Rigid Bodies: A Systematic Approach*](https://books.google.com/books?id=2y9e6BjxZf4C&pg=PA3). Cambridge University Press. p. 3. [ISBN](/source/ISBN_(identifier)) [978-0-521-85811-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-85811-3). Retrieved 2024-09-08.

1. ^ [***a***](#cite_ref-Teodorescu_4-0) [***b***](#cite_ref-Teodorescu_4-1) Teodorescu, Petre P. (2007-06-06). [*Mechanical Systems, Classical Models: Volume 1: Particle Mechanics*](https://books.google.com/books?id=k4H2AjWh9qQC&dq=%2522free+vector%2522+bound+vector&pg=PA5). Springer Science & Business Media. [ISBN](/source/ISBN_(identifier)) [978-1-4020-5442-6](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4020-5442-6).

1. **[^](#cite_ref-p822_5-0)** Merches, I.; Radu, D. (2014). [*Analytical Mechanics: Solutions to Problems in Classical Physics*](https://books.google.com/books?id=e9XMBQAAQBAJ&pg=PA379). CRC Press. p. 379. [ISBN](/source/ISBN_(identifier)) [978-1-4822-3940-9](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4822-3940-9). Retrieved 2024-09-09.

1. **[^](#cite_ref-z733_6-0)** Borisenko, A.I.; Tarapov, I.E.; Silverman, R.A. (2012). [*Vector and Tensor Analysis with Applications*](https://books.google.com/books?id=8eO7AQAAQBAJ&pg=PA2). Dover Books on Mathematics. Dover Publications. p. 2. [ISBN](/source/ISBN_(identifier)) [978-0-486-13190-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-13190-0). Retrieved 2024-09-08.

1. **[^](#cite_ref-p422_7-0)** "Appendix A. Linear Algebra from a Geometric Point of View". *Differential Geometry: A Geometric Introduction*. Ithaca, NY: David W. Henderson. 2013. pp. 121–138. [doi](/source/Doi_(identifier)):[10.3792/euclid/9781429799843-13](https://doi.org/10.3792%2Feuclid%2F9781429799843-13). [ISBN](/source/ISBN_(identifier)) [978-1-4297-9984-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4297-9984-3).

1. ^ [***a***](#cite_ref-w531_8-0) [***b***](#cite_ref-w531_8-1) ["ISO 80000-2:2019 - Quantities and units - Part 2: Mathematics"](https://www.iso.org/standard/64973.html). *ISO*. 2013-08-20. Retrieved 2024-09-08.

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