is the angle between and . Cross product The cross product or vector product is a binary operation on two vectors in three-dimensional space and is denoted by the symbol ×. The cross product A × B of the vectors A and B is a vector that is perpendicular to both and therefore normal to the plane containing them. It has many applications in mathematics, physics, and engineering. In function language, the cross product is a function . The components of the cross product are , and can also be written in components, using Einstein summation convention as where is the Levi-Civita symbol. It has the property that . Its magnitude is related to the angle between and by the identity The space and product form an algebra over a field, which is not commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket. Specifically, the space together with the product, is isomorphic to the Lie algebra of three-dimensional rotations, denoted . In order to satisfy the axioms of a Lie algebra, instead of associativity the cross product satisfies the Jacobi identity. For any three vectors and One can in n dimensions take the product of vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions. Abstract description It can be useful to describe three-dimensional space as a three-dimensional vector space over the real numbers. This differs from in a subtle way. By definition, there exists a basis for . This corresponds to an isomorphism between and : the construction for the isomorphism is found here. However, there is no 'preferred' or 'canonical basis' for . On the other hand, there is a preferred basis for , which is due to its description as a Cartesian product of copies of , that is, . This allows the definition of canonical projections, , where . For example, . This then allows the definition of the standard basis defined by where is the Kronecker delta. Written out in full, the standard basis is Therefore can be viewed as the abstract vector space, together with the additional structure of a choice of basis. Conversely, can be obtained by starting with and 'forgetting' the Cartesian product structure, or equivalently the standard choice of basis. As opposed to