In thenatural sciences, avector quantity (also known as avector physical quantity,physical vector, or simplyvector) is avector-valuedphysical quantity.[1][2]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.[3]Abound vector is defined as the combination of an ordinary vector quantity and apoint of application orpoint of action.[1][4]Bound vector quantities are formulated as adirected line segment, with a definite initial point besides the magnitude and direction of the main vector.[1][3]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).[5][6][4]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).[7][8]Asliding 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).Afree 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.[8]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.
