In physics and engineering, a vector is a physical quantity having both a magnitude and a direction, as opposed to a scalar, which only has a magnitude (and possibly a sign). Examples of vectors are the velocity of a particle, or the value of the electric field at a given point of space.
There are basic operations which can be performed on vectors: one can take the sum or difference of two vectors with the same physical dimensions, obtaining another vector; the product of a vector and a scalar, obtaining a vector with the same direction, and whose magnitude is the product of that of the original vector and the scalar; the dot product of two vectors, obtaining a scalar; and the cross product of two vectors, obtaining a pseudovector. Given a vector field (that is a vector-valued quantity defined at each point of space), differential operators such as the divergence (returning a scalar field), and the curl (returning a pseudovector field) can be defined.
Mathematically, vectors used in classical mechanics can be treated as elements of a three-dimensional Euclidean vector space, but in more advanced physics generalizations are used; for example, four-vectors in special relativity are elements of a (3 + 1)-dimensional Minkowski space. In curved spacetime of general relativity, the distinction between covariant and contravariant vectors becomes relevant. Vectors are an instance of the more general concept of a tensor: vectors are tensors of order 1, whereas scalars are tensors of order 0.
In mathematics, the term vector has a more general meaning, that of an element of a vector space, that is any set of elements which can be added together or multiplied by scalars satisfying certain properties; this meaning is something used for concepts in physics, such as a state vector, which don't necessarily have a direct relationship with either three-dimensional physical space or spacetime. See Vector (disambiguation) for those meanings; this article specifically deals with tensors of order 1.