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As mentioned above, the cross product can be interpreted as the exterior product in three dimensions by using the Hodge star operator to map 2-vectors to vectors. The Hodge dual of the exterior product yields an ( n − 2) -vector, which is a natural generalization of the cross product in any number of dimensions.
In two dimensions, the Levi-Civita symbol is defined by: = {+ (,) = (,) (,) = (,) = The values can be arranged into a 2 × 2 antisymmetric matrix: = (). Use of the two-dimensional symbol is common in condensed matter, and in certain specialized high-energy topics like supersymmetry [1] and twistor theory, [2] where it appears in the context of 2-spinors.
To paraphrase Baylis: Given two polar vectors (that is, true vectors) a and b in three dimensions, the cross product composed from a and b is the vector normal to their plane given by c = a × b. Given a set of right-handed orthonormal basis vectors { e ℓ}, the cross product is expressed in terms of its components as:
The Hodge star relates the exterior and cross product in three dimensions: [2] = =. Applied to three dimensions, the Hodge star provides an isomorphism between axial vectors and bivectors , so each axial vector a is associated with a bivector A and vice versa, that is: [ 2 ] A = ⋆ a , a = ⋆ A {\displaystyle \mathbf {A} ={\star }\mathbf {a ...
The restriction to 0, 1, 3 and 7 dimensions is related to Hurwitz's theorem, that normed division algebras are only possible in 1, 2, 4 and 8 dimensions. The cross product is formed from the product of the normed division algebra by restricting it to the 0, 1, 3, or 7 imaginary dimensions of the algebra, giving nonzero products in only three ...
In mathematics and physics, the right-hand rule is a convention and a mnemonic, utilized to define the orientation of axes in three-dimensional space and to determine the direction of the cross product of two vectors, as well as to establish the direction of the force on a current-carrying conductor in a magnetic field.
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.
The exterior product of two vectors can be identified with the signed area enclosed by a parallelogram the sides of which are the vectors. The cross product of two vectors in dimensions with positive-definite quadratic form is closely related to their exterior product.