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Exterior calc triple product

The three vectors spanning a parallelepiped have triple product equal to its volume.

In exterior algebra and geometric algebra the exterior product of two vectors is a bivector, while the exterior product of three vectors is a trivector. A bivector is an oriented plane element and a trivector is an oriented volume element, in the same way that a vector is an oriented line element. Given vectors a, b and c, the product

$ \mathbf{a} \wedge \mathbf{b} \wedge \mathbf{c} $

is a trivector with magnitude equal to the scalar triple product, and is the Hodge dual of the vector triple product. As the exterior product is associative brackets are not needed as it does not matter which of ab or bc is calculated first, though the order of the vectors in the product does matter. Geometrically the trivector abc corresponds to the parallelepiped spanned by a, b, and c, with bivectors ab, bc and ac matching the parallelogram faces of the parallelepiped.


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