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The cross product with respect to a right-handed coordinate system. In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol .
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
There are two lists of mathematical identities related to vectors: Vector algebra relations — regarding operations on individual vectors such as dot product, cross product, etc. Vector calculus identities — regarding operations on vector fields such as divergence, gradient, curl, etc.
As an example, the geometric product of two vectors = + = + since = and = and = , for other than and . A multivector A {\displaystyle A} may also be decomposed into even and odd components, which may respectively be expressed as the sum of the even and the sum of the odd grade components above:
The cross product of two vectors u and v would be represented as: By some conventions (e.g. in France and in some areas of higher mathematics), this is also denoted by a wedge, [ 12 ] which avoids confusion with the wedge product since the two are functionally equivalent in three dimensions: u ∧ v {\displaystyle \mathbf {u} \wedge \mathbf {v} }
The dot product takes in two vectors and returns a scalar, while the cross product [a] returns a pseudovector. Both of these have various significant geometric interpretations and are widely used in mathematics, physics, and engineering. The dyadic product takes in two vectors and returns a second order tensor called a dyadic in this context. A ...
In Euclidean 3-space, the wedge product has the same magnitude as the cross product (the area of the parallelogram formed by sides and ) but generalizes to arbitrary affine spaces and products between more than two vectors. Tensor product – for two vectors and , where and are vector spaces, their tensor product belongs to the tensor product ...
The seven-dimensional cross product is one way of generalizing the cross product to other than three dimensions, and it is the only other bilinear product of two vectors that is vector-valued, orthogonal, and has the same magnitude as in the 3D case. [2]