In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. A vector can be thought of as an arrow in Euclidean space, drawn from an initial point A pointing to a terminal point B. This vector is commonly denoted by
indicating that the arrow points from A to B. In this way, the arrow holds all the information of the vector quantity — the magnitude is represented by the arrow's length and the direction by the direction of the arrow's head and body. This magnitude and direction are those necessary to carry one from A to B. Indeed in Latin the word vector means "one who carries"; here the vector is what would carry a point from A to B.
Vectors have a variety of algebraic properties. Vectors may be scaled by stretching them out, or compressing them. They can be flipped around so as to point in the opposite direction. Two vectors sharing the same initial point can be added or subtracted according to the parallelogram law. Often the initial point of a vector is fixed at the origin of the Euclidean space, in which case the ensemble of vectors forms a vector space.
Vectors are fundamental in the physical sciences. A common example is velocity. Another is force — it has a magnitude and a direction and multiple forces sum according to the parallelogram law. Vectors also describe many other physical properties, such as displacement, acceleration, electric field, momentum, and angular momentum.
Spatial vectors are often described in terms of a particular frame of reference. Physically, this corresponds to an observer locating the endpoints of a vector with respect to a fixed system of measuring rods. A vector is then identified with a list of numbers, called the components of the vector, corresponding to the individual measurements along each rod (as a row vector or column vector). In Newtonian mechanics, a vector can be formally characterized in terms of how its components change (or transform) when an observer uses different Galilean reference frames.
In mathematics and physics, the concept of a spatial vector has been generalized in several ways. Vectors in four dimensions, called four-vectors, play a central role in the theory of relativity. Vectors which vary from point to point are called vector fields and are the basic object of study in vector calculus. In differential geometry, vectors can also be defined on a differentiable manifold as an element of the tangent space. Vectors may also be viewed as a certain type of tensor.