Triple product

In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product.
Scalar triple product.
The scalar triple product (also called the mixed product, box product, or triple scalar product) is defined as the dot product of one of the vectors with the cross product of the other two.
Geometric interpretation.
Geometrically, the scalar triple product
is the (signed) volume of the parallelepiped defined by the three vectors given. Here, the parentheses may be omitted without causing ambiguity, since the dot product cannot be evaluated first. If it were, it would leave the cross product of a scalar and a vector, which is not defined.
Scalar or pseudoscalar.
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 must be pseudoscalar-valued.
If T is a rotation operator, then
but if T is an improper rotation, then
As an exterior product.
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, 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 or 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 , and matching the parallelogram faces of the parallelepiped.
As a trilinear functional.
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.
Vector triple product.
The vector triple product is defined as the cross product of one vector with the cross product of the other two. The following relationship holds:
This is known as triple product expansion, or Lagrange's formula, 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. A proof is provided below. Some textbooks write the identity as formula_12 such that a more familiar mnemonic is obtained "BAC − CAB", as in “back of the cab”.
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. Another useful formula follows:
These formulas are very useful in simplifying vector calculations in physics. A related identity regarding gradients and useful in vector calculus is Lagrange's formula of vector cross-product identity:
This can be also regarded as a special case of the more general Laplace–de Rham operator formula_17.
Proof.
The formula_18 component of formula_19 is given by:
Similarly, the formula_21 and formula_22 components of formula_23 are given by:
By combining these three components we obtain:
Using geometric algebra.
If geometric algebra is used the cross product b × c of vectors is expressed as their exterior product b∧c, a bivector. The second cross product cannot be expressed as an exterior product, otherwise the scalar triple product would result. Instead a left contraction can be used, so the formula becomes
The proof follows from the properties of the contraction. The result is the same vector as calculated using a × (b × c).
Interpretations.
Tensor calculus.
In tensor notation the triple product is expressed using the Levi-Civita symbol:
and
referring to the formula_29th component of the resulting vector. This can be simplified by performing a contraction on the Levi-Civita symbols, 
formula_30
where formula_31 if formula_32 and formula_33 if formula_34. We can reason out this identity by recognizing that the index formula_35 will be summed out leaving only formula_29 and formula_37. In the first term, we fix formula_38 and thus formula_39. Likewise, in the second term, we fix formula_40 and thus formula_41.
Returning to the triple cross product,
Vector calculus.
Consider the flux integral of the vector field formula_43 across the parametrically-defined surface formula_44: formula_45. The unit normal vector formula_46 to the surface is given by formula_47, so the integrand formula_48 is a scalar triple product.