A spatial<unk>  or simply<unk>  is a geometric object which has both a magnitude and a direction.  A<unk> is frequently represented by a line segment connecting the initial point A with the  terminal point B and denoted
The magnitude is the length of the segment and the direction characterizes the displacement of B relative to A: how much one should move the point A to "carry" it to the point B.
Many algebraic operations on real numbers have close analogues for<unk> . <unk>  can be added, subtracted, multiplied by a number, and flipped around so that the direction is reversed. These operations obey the familiar algebraic laws: commutativity, associativity, distributivity. The sum of two<unk>  with the same initial point can be found geometrically using the parallelogram law. Multiplication by a positive number, commonly called a scalar in this context, amounts to changing the magnitude of<unk>  that is, stretching or compressing it while keeping its direction; multiplication by -1 preserves the magnitude of the<unk> but reverses its direction.
Cartesian coordinates provide a systematic way of describing<unk>  and operations on them. A<unk> becomes a triple of real numbers, its scalar components. Addition of<unk>  and multiplication of a<unk> by a scalar are simply done component by component, see coordinate<unk> <unk>  play an important role in physics: velocity and acceleration of a moving object and forces acting on a body are all described by<unk> . Many other physical quantities can be usefully thought of as<unk> . One has to keep in mind, however, that the components of a physical<unk> depend on the coordinate system used to describe it. Other<unk> like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors.
Informally, a<unk> is a quantity characterized by a magnitude (in mathematics a number, in physics a number times a unit) and a direction, often represented graphically by an arrow.  Sometimes, one speaks of bound or fixed<unk> , which are<unk>  whose initial point is the origin.  This is in contrast to free<unk> , which are<unk>  whose initial point is not necessarily the origin.<unk>  are fundamental in the physical sciences.  They can be used to represent any quantity that has both a magnitude and direction, such as velocity, the magnitude of which is speed. For example, the velocity 5 meters per second upward could be represented by the<unk> (0,5). Another quantity represented by a<unk> is force, since it has a magnitude and direction.<unk>  also describe many other physical quantities, such as displacement, acceleration, electric and magnetic fields, momentum, and angular momentum.
In the Cartesian coordinate system, a<unk> can be represented by identifying the coordinates of its initial and terminal point.  For instance, the points A = (1,0,0) and B = (0,1,0) in space determine the free<unk> '"`UNIQ--postMath-00000002-QINU`"' pointing from the point x=1 on the x-axis to the point y=1 on the y-axis.
Typically in Cartesian coordinates, one considers primarily bound<unk> .  A bound<unk> is determined by the coordinates of the terminal point, its initial point always having the coordinates of the origin O = (0,0,0).  Thus the bound<unk> represented by (1,0,0) is a<unk> of unit length pointing from the origin up the positive x-axis.
The coordinate representation of<unk>  allows the algebraic features of<unk>  to be expressed in a convenient numerical fashion.  For example, the sum of the<unk>  (1,2,3) and (-2,0,4) is the<unk> In the geometrical and physical settings, sometimes it is possible to associate, in a natural way, a length to<unk>  as well as the notion of an angle between two<unk> .  When the length of<unk>  is defined, it is possible to also define a dot product — a scalar-valued product of two<unk>  — which gives a convenient algebraic characterization of both length and angle.  In three-dimensions, it is further possible to define a cross product which supplies an algebraic characterization of area.
However, it is not always possible or desirable to define the length of a<unk> in a natural way.  This more general type of spatial<unk> is the subject of<unk> spaces (for bound<unk>  and affine spaces (for free<unk> .
In more general sorts of coordinate systems, rotations of a<unk> (and also of tensors) can be generalized and categorized to admit an analogous characterization by their covariance and contravariance under changes of coordinates.
In mathematics, a<unk> is considered more than a representation of a physical quantity. In general, a<unk> is any element of a<unk> space over some field. The spatial<unk>  of this article are a very special case of this general definition (they are not simply any element of Rd in d dimensions), which includes a variety of mathematical objects (algebras, the set of all functions from a given domain to a given linear range, and linear transformations). Note that under this definition, a tensor is a special<unk>
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