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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 ×: R 3 × R 3 → R 3 {\displaystyle \times :\mathbb {R} ^{3}\times \mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}}.

The components of the cross product are A × B = {\displaystyle \mathbf {A} \times \mathbf {B} =}, and can also be written in components, using Einstein summation convention as (A × B) i = ε i j k A j B k {\displaystyle (\mathbf {A} \times \mathbf {B})_{i}=\varepsilon _{ijk}A_{j}B_{k}} where ε i j k {\displaystyle \varepsilon _{ijk}} is the Levi-Civita symbol. It has the property that A × B = − B × A {\displaystyle \mathbf {A} \times \mathbf {B} =-\mathbf {B} \times \mathbf {A} }.

Its magnitude is related to the angle θ {\displaystyle \theta } between A {\displaystyle \mathbf {A} } and B {\displaystyle \mathbf {B} } 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, (R 3, ×) {\displaystyle (\mathbb {R} ^{3},\times)} is isomorphic to the Lie algebra of three-dimensional rotations, denoted s o (3) {\displaystyle {\mathfrak {so}}(3)}. In order to satisfy the axioms of a Lie algebra, instead of associativity the cross product satisfies the Jacobi identity. For any three vectors A, B {\displaystyle \mathbf {A},\mathbf {B} } and C {\displaystyle \mathbf {C} }

One can in n dimensions take the product of n − 1 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.