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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 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} }
As the Levi-Civita symbol is a pseudotensor, the result of taking a cross product is a pseudovector, not a vector. [ 5 ] Under a general coordinate change , the components of the permutation tensor are multiplied by the Jacobian of the transformation matrix .
In three dimensions, the cross product of two vectors with respect to a positively oriented orthonormal basis, meaning that =, can be expressed as: = Here, ε j k i = ε i j k {\displaystyle \varepsilon _{\,jk}^{i}=\varepsilon _{ijk}} is the Levi-Civita 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.
Right-hand rule for cross product. The cross product of vectors and is a vector perpendicular to the plane spanned by and with the direction given by the right-hand rule: If you put the index of your right hand on and the middle finger on , then the thumb points in the direction of .
2.7 Cross product rule. 3 Second derivative identities. ... where = ±1 or 0 is the Levi-Civita parity symbol. For a tensor field of order k > 1, the tensor ...
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