Inmathematics andphysics, avector is aphysical quantity that cannot be expressed by a single number (ascalar). The term may also be used to refer to elements of somevector spaces, and in some contexts, is used fortuples, which arefinite sequences (of numbers or other objects) of a fixed length.
Historically, vectors were introduced ingeometry and physics (typically inmechanics) for quantities that have both a magnitude and a direction, such asdisplacements,forces andvelocity. Such quantities are represented bygeometric vectors in the same way asdistances,masses andtime are represented byreal numbers.
Both geometric vectors and tuples can be added and scaled, and these vector operations led to the concept of a vector space, which is aset equipped with avector addition and ascalar multiplication that satisfy someaxioms generalizing the main properties of operations on the above sorts of vectors. A vector space formed by geometric vectors is called aEuclidean vector space, and a vector space formed by tuples is called acoordinate vector space.
Many vector spaces are considered in mathematics, such asextension fields,polynomial rings,algebras andfunction spaces. The termvector is generally not used for elements of these vector spaces, and is generally reserved for geometric vectors, tuples, and elements of unspecified vector spaces (for example, when discussing general properties of vector spaces).
Inmathematics,physics, andengineering, aEuclidean vector or simply a vector (sometimes called a geometric vector[1] or spatial vector[2]) is a geometric object that hasmagnitude (orlength) anddirection. Euclidean vectors can be added and scaled to form avector space. Avector quantity is a vector-valuedphysical quantity, includingunits of measurement and possibly asupport, formulated as adirected line segment. A vector is frequently depicted graphically as an arrow connecting aninitial pointA with aterminal pointB,[3] and denoted by
A vector is what is needed to "carry" the pointA to the pointB; the Latin wordvector means 'carrier'.[4] It was first used by 18th centuryastronomers investigating planetary revolution around the Sun.[5] The magnitude of the vector is the distance between the two points, and the direction refers to the direction ofdisplacement fromA toB. Manyalgebraic operations onreal numbers such asaddition,subtraction,multiplication, andnegation have close analogues for vectors,[6] operations which obey the familiar algebraic laws ofcommutativity,associativity, anddistributivity. These operations and associated laws qualifyEuclidean vectors as an example of the more generalized concept of vectors defined simply as elements of avector space.
Vectors play an important role inphysics: thevelocity andacceleration of a moving object and theforces acting on it can all be described with vectors.[7] Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example,position ordisplacement), 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 thecoordinate system used to describe it. Other vector-likemathematical objects that describephysical quantities, such aspseudovectors andtensors, transform in a similar way under changes of the coordinate system.[8]In thenatural sciences, avector quantity (also known as a vector physical quantity, physical vector, or simply vector) is a vector-valuedphysical quantity.[9][10]It is typically formulated as the product of aunit of measurement and avectornumerical value (unitless), often aEuclidean vector withmagnitude anddirection.For example, aposition vector inphysical space may be expressed asthreeCartesian coordinates withSI unit ofmeters.
Inphysics andengineering, particularly inmechanics, a physical vector may be endowed with additional structure compared to a geometrical vector.[11]A bound vector is defined as the combination of an ordinary vector quantity and apoint of application orpoint of action.[9][12]Bound vector quantities are formulated as adirected line segment, with a definite initial point besides the magnitude and direction of the main vector.[9][11]For example, aforce on theEuclidean plane has two Cartesian components in SI unit ofnewtons and an accompanying two-dimensional position vector in meters, for a total of four numbers on the plane (and six in space).[13][14][12]A simpler example of a bound vector is thetranslation vector from an initial point to an end point; in this case, the bound vector is anordered pair of points in the same position space, with all coordinates having the samequantity dimension and unit (length and meters).[15][16]A sliding vector is the combination of an ordinary vector quantity and aline of application orline of action, over which the vector quantity can be translated (without rotations).A free vector is a vector quantity having an undefinedsupport or region of application; it can be freely translated with no consequences; adisplacement 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 ofmetric.For example, an event inspacetime may be represented as aposition four-vector, withcoherent derived unit of meters: it includes a position Euclidean vector and atimelike component,t⋅c0 (involving thespeed of light).In that case, theMinkowski metric is adopted instead of theEuclidean metric.
Vector quantities are a generalization ofscalar quantities and can be further generalized astensor quantities.[16]Individual vectors may be ordered in asequence over time (atime series), such as position vectorsdiscretizing atrajectory.A vector may also result from theevaluation, at a particular instant, of a continuousvector-valued function (e.g., thependulum equation).In the natural sciences, the term "vector quantity" also encompassesvector fields defined over atwo- or three-dimensionalregion of space, such aswind velocity over Earth's surface.
Pseudo vectors andbivectors are also admitted as physical vector quantities.
Inmathematics andphysics, avector space (also called a linear space) is aset whose elements, often calledvectors, can be added together and multiplied ("scaled") by numbers calledscalars. The operations of vector addition andscalar multiplication must satisfy certain requirements, calledvectoraxioms. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars:real numbers andcomplex numbers. Scalars can also be, more generally, elements of anyfield.
Vector spaces generalizeEuclidean vectors, which allow modeling ofphysical quantities (such asforces andvelocity) that have not only amagnitude, but also adirection. The concept of vector spaces is fundamental forlinear algebra, together with the concept ofmatrices, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studyingsystems of linear equations.
Vector spaces are characterized by theirdimension, which, roughly speaking, specifies the number of independent directions in the space. This means that for two vector spaces over a given field and with the same dimension, the properties that depend only on the vector-space structure are exactly the same (technically, the vector spaces areisomorphic). A vector space isfinite-dimensional if its dimension is anatural number. Otherwise, it isinfinite-dimensional, and its dimension is aninfinite cardinal. Finite-dimensional vector spaces occur naturally ingeometry and related areas. Infinite-dimensional vector spaces occur in many areas of mathematics. For example,polynomial rings arecountably infinite-dimensional vector spaces, and manyfunction spaces have thecardinality of the continuum as a dimension.
Many vector spaces that are considered in mathematics are also endowed with otherstructures. This is the case ofalgebras, which includefield extensions, polynomial rings,associative algebras andLie algebras. This is also the case oftopological vector spaces, which include function spaces,inner product spaces,normed spaces,Hilbert spaces andBanach spaces.Everyalgebra over a field is a vector space, but elements of an algebra are generally not called vectors. However, in some cases, they are calledvectors, mainly due to historical reasons.
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The set oftuples ofn real numbers has a natural structure of vector space defined by component-wise addition andscalar multiplication. It is common to call these tuplesvectors, even in contexts where vector-space operations do not apply. More generally, when some data can be represented naturally by vectors, they are often calledvectors even when addition and scalar multiplication of vectors are not valid operations on these data.[disputed –discuss] Here are some examples.
Calculus serves as a foundational mathematical tool in the realm of vectors, offering a framework for the analysis and manipulation of vector quantities in diverse scientific disciplines, notablyphysics andengineering. Vector-valued functions, where the output is a vector, are scrutinized using calculus to derive essential insights into motion within three-dimensional space. Vector calculus extends traditional calculus principles to vector fields, introducing operations likegradient,divergence, andcurl, which find applications in physics and engineering contexts.Line integrals, crucial for calculating work along a path within force fields, andsurface integrals, employed to determine quantities likeflux, illustrate the practical utility of calculus in vector analysis.Volume integrals, essential for computations involving scalar or vector fields over three-dimensional regions, contribute to understandingmass distribution,charge density, and fluid flow rates.[citation needed]
Avector field is avector-valued function that, generally, has a domain of the same dimension (as amanifold) as its codomain,
