Vector quantity: Difference between revisions

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{{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.br/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.br/books?id=k4H2AjWh9qQC&pg=PA5&dq=%2522free+vector%2522+bound+vector&hlpg=en&newbks=1&newbks_redir=0&sa=X&redir_esc=yPA5 |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]] and an accompanying two-dimensional position vector in meters, for a total of four numbers on the plane (and six in space).<ref name="p822">{{cite book | lastlast1=Merches | firstfirst1=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.br/books?id=e9XMBQAAQBAJ&pg=PA379 | access-date=2024-09-09 | page=379}}</ref><ref name="z733">{{cite book | lastlast1=Borisenko | firstfirst1=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/>

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 an 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 | pagepages=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 [[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.