Editing Euclidean vector (section)

{{Short description|Geometric object that has length and direction}}
{{For-text|mathematical vectors in general|[[Vector (mathematics and physics)]]|other uses|[[Vector (disambiguation)]]}}
[[File:Vector from A to B.svg|class=skin-invert-image|thumb|A vector <math display=inline>\stackrel \rightarrow{a}</math> pointing from point ''A'' to point ''B'']]

In [[mathematics]], [[physics]], and [[engineering]], a '''Euclidean vector''' or simply a '''vector''' (sometimes called a '''geometric vector'''<ref>{{harvnb|Ivanov|2001}}</ref> or '''spatial vector'''<ref>{{harvnb|Heinbockel|2001}}</ref>) is a geometric object that has [[Magnitude (mathematics)|magnitude]] (or [[Euclidean norm|length]]) and [[Direction (geometry)|direction]]. Euclidean vectors can be added and scaled to form a [[vector space]]. A ''[[vector quantity]]'' is a vector-valued [[physical quantity]], including [[units of measurement]] and possibly a [[support (mathematics)|support]], formulated as a ''[[directed line segment]]''. A vector is frequently depicted graphically as an arrow connecting an ''initial point'' ''A'' with a ''terminal point'' ''B'',<ref>{{harvnb|Itô|1993|p=1678}}; {{harvnb|Pedoe|1988}}</ref> and denoted by <math display=inline>\stackrel \longrightarrow{AB}.</math>

A vector is what is needed to "carry" the point ''A'' to the point ''B''; the Latin word {{lang|la|vector}} means 'carrier'.<ref>Latin: {{lang|la|vectus}}, [[perfect participle]] of {{lang|la|vehere}}, 'to carry', {{lang|la|veho}} = 'I carry'.  For historical development of the word ''vector'', see {{OED|vector ''n.''}} and {{cite web|author = Jeff Miller| url = http://jeff560.tripod.com/v.html | title = Earliest Known Uses of Some of the Words of Mathematics | access-date = 2007-05-25}}</ref> It was first used by 18th century [[astronomers]] investigating planetary revolution around the Sun.<ref>{{cite book|title=The Oxford English Dictionary.|year=2001|publisher=Clarendon Press|location=London|isbn=9780195219425|edition=2nd.}}</ref> The magnitude of the vector is the distance between the two points, and the direction refers to the direction of [[Displacement (geometry)|displacement]] from ''A'' to ''B''. Many [[algebraic operation]]s on [[real number]]s such as [[addition]], [[subtraction]], [[multiplication]], and [[Additive inverse|negation]] have close analogues for vectors,<ref name=":1">{{Cite web|title=vector {{!}} Definition & Facts|url=https://www.britannica.com/science/vector-mathematics|access-date=2020-08-19|website=Encyclopedia Britannica|language=en}}</ref> operations which obey the familiar algebraic laws of [[commutativity]], [[associativity]], and [[distributivity]]. These operations and associated laws qualify [[Euclidean space|Euclidean]] vectors as an example of the more generalized concept of vectors defined simply as elements of a [[vector space]].

Vectors play an important role in [[physics]]: the [[velocity]] and [[acceleration]] of a moving object and the [[force]]s acting on it can all be described with vectors.<ref name=":2">{{Cite web|title=Vectors|url=https://www.mathsisfun.com/algebra/vectors.html|access-date=2020-08-19|website=www.mathsisfun.com}}</ref> Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, [[position (vector)|position]] or [[displacement (vector)|displacement]]), their magnitude and direction can still be represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the [[coordinate system]] used to describe it. Other vector-like objects that describe [[physical quantities]] and transform in a similar way under changes of the coordinate system include [[pseudovector]]s and [[tensor]]s.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Vector|url=https://mathworld.wolfram.com/Vector.html|access-date=2020-08-19|website=mathworld.wolfram.com|language=en}}</ref>