In mathematics, the cross product is a binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which is perpendicular to the two input vectors. By contrast, the dot product produces a scalar result. In many engineering and physics problems, it is handy to be able to construct a perpendicular vector from two existing vectors, and the cross product provides a means for doing so. The cross product is also known as the Gibbs vector product.
The cross product is not defined except in three-dimensions (and the algebra defined by the cross product is not associative.) Like the dot product, it depends on the metric of Euclidean space. Unlike the dot product, it also depends on the choice of orientation or "handedness." Certain features of the cross product can be generalized to other situations. For arbitrary choices of orientation, the cross product must be regarded, not as a vector, but as a pseudovector. For arbitrary choices of metric, and in arbitrary dimensions, the cross product can be generalized by the exterior product of vectors, defining a two-form instead of a vector.