From Glossary of Meteorology
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Any quantity, such as force, velocity, or acceleration, that has both magnitude and direction at each point in space, as opposed to a scalar that has magnitude only.

Such a quantity may be represented geometrically by an arrow of length proportional to its magnitude, pointing in the assigned direction. A unit vector is a vector of unit length; in particular, the three unit vectors along the positive x, y, and z axes of rectangular Cartesian coordinates are denoted, respectively, by i, j, and k. Any vector A can be represented in terms of its components a1, a2, and a3 along the coordinate axes x, y, and z, respectively; for example, A = a1i + a2j + a3k. A vector drawn from a fixed origin to a given point (x, y, z) is called a position vector and is usually symbolized by r; in rectangular Cartesian coordinates,
Equations written in vector form are valid in any coordinate system. Mathematically, a vector is a single-row or single-column array of functions obeying certain laws of transformation.
See scalar product, vector product, Helmholtz's theorem.