{{Short description|Physical quantity that is a vector}} In the [[natural sciences]], a '''vector quantity''' (also known as a '''vector physical quantity''', '''physical vector''', or simply '''vector''') is a [[vector (mathematics and physics)|vector]]-valued [[physical quantity]].<ref name="a306">{{cite web | title=Details for IEV number 102-03-21: "vector quantity" | website=International Electrotechnical Vocabulary | url=https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=102-03-21 | language=ja | access-date=2024-09-07}}</ref><ref name="o531">{{cite web | title=Details for IEV number 102-03-04: "vector" | website=International Electrotechnical Vocabulary | url=https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=102-03-04 | language=ja | access-date=2024-09-07}}</ref> It is typically formulated as the product of a ''[[unit of measurement]]'' and a ''vector [[numerical value]]'' ([[unitless]]), often a [[Euclidean vector]] with [[vector norm|magnitude]] and [[direction (geometry)|direction]]. For example, a [[position vector]] in [[physical space]] may be expressed as [[three dimensional|three]] [[Cartesian coordinates]] with [[SI unit]] of [[meters]].
In [[physics]] and [[engineering]], particularly in [[mechanics]], a physical vector may be endowed with additional structure compared to a geometrical vector.<ref name="m813">{{cite book | last=Rao | first=A. | title=Dynamics of Particles and Rigid Bodies: A Systematic Approach | publisher=Cambridge University Press | year=2006 | isbn=978-0-521-85811-3 | url=https://books.google.com/books?id=2y9e6BjxZf4C&pg=PA3 | access-date=2024-09-08 | page=3}}</ref> A '''bound vector''' is defined as the combination of an ordinary vector quantity and a ''[[point of application]]'' or ''point of action''.<ref name="a306"/> <ref name=Teodorescu>{{Cite book |last=Teodorescu |first=Petre P. |url=https://books.google.com/books?id=k4H2AjWh9qQC&dq=%2522free+vector%2522+bound+vector&pg=PA5 |title=Mechanical Systems, Classical Models: Volume 1: Particle Mechanics |date=2007-06-06 |publisher=Springer Science & Business Media |isbn=978-1-4020-5442-6 |language=en}}</ref> Bound vector quantities are formulated as a ''[[directed line segment]]'', with a definite initial point besides the magnitude and direction of the main vector.<ref name="a306"/><ref name="m813"/> For example, a [[force]] on the [[Euclidean plane]] has two Cartesian components in SI unit of [[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).<ref name="p822">{{cite book | last1=Merches | first1=I. | last2=Radu | first2=D. | title=Analytical Mechanics: Solutions to Problems in Classical Physics | publisher=CRC Press | year=2014 | isbn=978-1-4822-3940-9 | url=https://books.google.com/books?id=e9XMBQAAQBAJ&pg=PA379 | access-date=2024-09-09 | page=379}}</ref><ref name="z733">{{cite book | last1=Borisenko | first1=A.I. | last2=Tarapov | first2=I.E. | last3=Silverman | first3=R.A. | title=Vector and Tensor Analysis with Applications | publisher=Dover Publications | series=Dover Books on Mathematics | year=2012 | isbn=978-0-486-13190-0 | url=https://books.google.com/books?id=8eO7AQAAQBAJ&pg=PA2 | access-date=2024-09-08 | page=2}}</ref><ref name=Teodorescu/> A simpler example of a bound vector is the [[translation (geometry)|translation]] vector from an initial point to an end point; in this case, the bound vector is an [[ordered pair]] of points in the same position space, with all coordinates having the same [[quantity dimension]] and unit (length and meters).<ref name="p422">{{cite book | title=Differential Geometry: A Geometric Introduction | chapter=Appendix A. Linear Algebra from a Geometric Point of View | publisher=David W. Henderson | publication-place=Ithaca, NY | date=2013 | isbn=978-1-4297-9984-3 | doi=10.3792/euclid/9781429799843-13 | pages=121–138}}</ref><ref name="w531"/> A '''sliding vector''' is the combination of an ordinary vector quantity and a ''[[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]] 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 [[distance metric|metric]]. 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, {{math|''t''{{sdot}}''c''<sub>0</sub>}} (involving the [[speed of light]]). In that case, the [[Minkowski metric]] is adopted instead of the [[Euclidean metric]].
Vector quantities are a generalization of [[scalar quantity|scalar quantities]] and can be further generalized as [[tensor quantity|tensor quantities]].<ref name="w531">{{cite web | title=ISO 80000-2:2019 - Quantities and units - Part 2: Mathematics | website=ISO | date=2013-08-20 | url=https://www.iso.org/standard/64973.html | access-date=2024-09-08}}</ref> Individual vectors may be ordered in a [[sequence]] over time (a [[time series]]), such as position vectors [[discretizing]] a [[trajectory]]. A vector may also result from the [[function evaluation|evaluation]], at a particular instant, of a continuous [[vector-valued function]] (e.g., the [[pendulum equation]]). In the natural sciences, the term "vector quantity" also encompasses ''[[vector field]]s'' defined over a [[Two-dimensional|two-]] or three-dimensional [[region (mathematics)|region]] of space, such as [[wind velocity]] over Earth's surface. [[Pseudo vector]]s and [[bivector]]s are also admitted as physical vector quantities.
==See also== *[[List of vector quantities]] *[[Vector representation]]
==References== {{reflist}}
{{physics-stub}} [[Category:Vector physical quantities|*]] [[Category:Vectors (mathematics and physics)]]