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In mathematics, the cross product or vector product is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space , and is denoted by the symbol × {\displaystyle \times }. Given two linearly independent vectors a and b, the cross product, a × b , is a vector that is perpendicular to both a and b, and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product.
If two vectors have the same direction or have the exact opposite direction from each other , or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths.
The cross product is anticommutative and is distributive over addition = a × b + a × c]. The space E {\displaystyle E} together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.
Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on a choice of orientation of the space. In connection with the cross product, the exterior product of vectors can be used in arbitrary dimensions and is independent of the orientation of the space.