Vector quadruple product

The vector quadruple product of four vectors A, B, C and D in three dimensions is defined as:

(mathbf{A} times mathbf{B}) cdot (mathbf{C} times mathbf{D}) = (mathbf{A} cdot mathbf{C})(mathbf{B} cdot mathbf{D}) - (mathbf{A} cdot mathbf{D})(mathbf{B} cdot mathbf{C})

where the "×" and "·" symbols denote respectively the cross product and dot product.

A quadruple product which is also a vector can be defined, which satisfies the following identities:

(mathbf{A} times mathbf{B}) times (mathbf{C} times mathbf{D}) = (mathbf{A} times mathbf{C})(mathbf{B} cdot mathbf{D})+(mathbf{B} times mathbf{D})(mathbf{A} cdot mathbf{C}) - (mathbf{A} times mathbf{D})(mathbf{B} cdot mathbf{C})-(mathbf{B} times mathbf{C})(mathbf{A} cdot mathbf{D})

and

(mathbf{A} times mathbf{B}) times (mathbf{C} times mathbf{D}) = [mathbf{A},mathbf{B}, mathbf{D}]mathbf{C}-[mathbf{A},mathbf{B}, mathbf{C}]mathbf{D}=

[mathbf{A},mathbf{C}, mathbf{D}]mathbf{B}-[mathbf{B}, mathbf{C},mathbf{D}]mathbf{A}

where $[mathbf{A},mathbf{B}, mathbf{C}]$ is the scalar triple product.

References

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Last updated on Thursday September 04, 2008 at 03:31:25 PDT (GMT -0700)
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