Vector quantity: Difference between revisions
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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> |
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#REDIRECT[[Euclidean vector]] |
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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]]. |
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For example, a [[position vector]] in [[physical space]] may be expressed as [[three dimensional|three]] [[Cartesian coordinates]] with [[SI unit]] of [[meters]]. |
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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.br/books?id=2y9e6BjxZf4C&pg=PA3 | access-date=2024-09-08 | page=3}}</ref> |
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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"/> |
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<ref name=Teodorescu>{{Cite book |last=Teodorescu |first=Petre P. |url=https://books.google.com.br/books?id=k4H2AjWh9qQC&pg=PA5&dq=%2522free+vector%2522+bound+vector&hl=en&newbks=1&newbks_redir=0&sa=X&redir_esc=y |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> |
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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"/> |
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For example, a [[force]] on the [[Euclidean plane]] has two Cartesian components in SI unit of [[newtons]] and an accompanying two-dimensional position vector in meters, for a total of four numbers on the plane.<ref name="z733">{{cite book | last=Borisenko | first=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.br/books?id=8eO7AQAAQBAJ&pg=PA2 | access-date=2024-09-08 | page=2}}</ref><ref name=Teodorescu/> |
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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). |
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A '''free vector''' is a vector quantity having an undefined [[Support (mathematics)|support]] or region of application; it can be freely translated with no consequences; a [[displacement vector]] is a prototypical example of free vector. |
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Aside from the notion of units and support, physical vector quantities may also differ from Euclidean vectors in terms of [[distance metric|metric]]. |
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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]]). |
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In that case, the [[Minkowski metric]] is adopted instead of the [[Euclidean metric]]. |
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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> |
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In the natural sciences, the term "vector quantity" also encompasses ''[[vector field]]s'', which are [[vector-valued function]]s over a [[region (mathematics)|region]] of space, such as [[wind velocity]] over Earth's surface. |
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[[Pseudo vector]]s and [[bivector]]s are also admitted as physical vector quantities. |
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==See also== |
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*[[List of vector quantities]] |
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==References== |
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{{reflist}} |
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[[Category:Physical quantities]] |
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[[Category:Vectors (mathematics and physics)]] |
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Revision as of 04:31, 8 September 2024
In the natural sciences, a vector quantity (also known as a vector physical quantity, physical vector, or simply vector) is a vector-valued physical quantity.[1][2] It is typically formulated as the product of a unit of measurement and a vector numerical value (unitless), often a Euclidean vector with magnitude and direction. For example, a position vector in physical space may be expressed as 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.[3] A bound vector is defined as the combination of an ordinary vector quantity and a point of application or point of action.[1] [4] Bound vector quantities are formulated as a directed line segment, with a definite initial point besides the magnitude and direction of the main vector.[1][3] For example, a force on the Euclidean plane has two Cartesian components in SI unit of newtons and an accompanying two-dimensional position vector in meters, for a total of four numbers on the plane.[5][4] 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 support 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 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, t ⋅ c0 (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 quantities and can be further generalized as tensor quantities.[6] In the natural sciences, the term "vector quantity" also encompasses vector fields, which are vector-valued functions over a region of space, such as wind velocity over Earth's surface. Pseudo vectors and bivectors are also admitted as physical vector quantities.
See also
References
- ^ a b c "Details for IEV number 102-03-21: "vector quantity"". International Electrotechnical Vocabulary (in Japanese). Retrieved 2024-09-07.
- ^ "Details for IEV number 102-03-04: "vector"". International Electrotechnical Vocabulary (in Japanese). Retrieved 2024-09-07.
- ^ a b Rao, A. (2006). Dynamics of Particles and Rigid Bodies: A Systematic Approach. Cambridge University Press. p. 3. ISBN 978-0-521-85811-3. Retrieved 2024-09-08.
- ^ a b Teodorescu, Petre P. (2007-06-06). Mechanical Systems, Classical Models: Volume 1: Particle Mechanics. Springer Science & Business Media. ISBN 978-1-4020-5442-6.
- ^ Borisenko, A.I.; Tarapov, I.E.; Silverman, R.A. (2012). Vector and Tensor Analysis with Applications. Dover Books on Mathematics. Dover Publications. p. 2. ISBN 978-0-486-13190-0. Retrieved 2024-09-08.
- ^ "ISO 80000-2:2019 - Quantities and units - Part 2: Mathematics". ISO. 2013-08-20. Retrieved 2024-09-08.