Triple product
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors.
Geometrically, the scalar triple product is the (signed) volume of the parallelepiped defined by the three vectors given.
Although the scalar triple product gives the volume of the parallelepiped, it is the signed volume, the sign depending on the orientation of the frame or the parity of the permutation of the vectors.
This means the product is negated if the orientation is reversed, for example by a parity transformation, and so is more properly described as a pseudoscalar if the orientation can change.
This also relates to the handedness of the cross product; the cross product transforms as a pseudovector under parity transformations and so is properly described as a pseudovector.
The dot product of two vectors is a scalar but the dot product of a pseudovector and a vector is a pseudoscalar, so the scalar triple product (of vectors) must be pseudoscalar-valued.
If T is a proper rotation then but if T is an improper rotation then Strictly speaking, a scalar does not change at all under a coordinate transformation.
(For example, the factor of 2 used for doubling a vector does not change if the vector is in spherical vs. rectangular coordinates.)
However, if each vector is transformed by a matrix then the triple product ends up being multiplied by the determinant of the transformation matrix, which could be quite arbitrary for a non-rotation.
That is, the triple product is more properly described as a scalar density.
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 is a trivector with magnitude equal to the scalar triple product, i.e. and is the Hodge dual of the scalar triple product.
As the exterior product is associative brackets are not needed as it does not matter which of a ∧ b or b ∧ c is calculated first, though the order of the vectors in the product does matter.
Geometrically the trivector a ∧ b ∧ c corresponds to the parallelepiped spanned by a, b, and c, with bivectors a ∧ b, b ∧ c and a ∧ c matching the parallelogram faces of the parallelepiped.
The triple product is identical to the volume form of the Euclidean 3-space applied to the vectors via interior product.
It also can be expressed as a contraction of vectors with a rank-3 tensor equivalent to the form (or a pseudotensor equivalent to the volume pseudoform); see below.
The following relationship holds: This is known as triple product expansion, or Lagrange's formula,[2][3] although the latter name is also used for several other formulas.
Its right hand side can be remembered by using the mnemonic "ACB − ABC", provided one keeps in mind which vectors are dotted together.
Some textbooks write the identity as
Since the cross product is anticommutative, this formula may also be written (up to permutation of the letters) as: From Lagrange's formula it follows that the vector triple product satisfies: which is the Jacobi identity for the cross product.
A related identity regarding gradients and useful in vector calculus is Lagrange's formula of vector cross-product identity:[4] This can be also regarded as a special case of the more general Laplace–de Rham operator
are given by: By combining these three components we obtain: If geometric algebra is used the cross product b × c of vectors is expressed as their exterior product b∧c, a bivector.
In tensor notation, the triple product is expressed using the Levi-Civita symbol:[8]
This can be simplified by performing a contraction on the Levi-Civita symbols,
is the generalized Kronecker delta function.
We can reason out this identity by recognizing that the index
Returning to the triple cross product,
Consider the flux integral of the vector field
The unit normal vector
