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.
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.
Geometrically, the scalar triple product
is the (signed) volume of the parallelepiped defined by the three vectors given.
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. That is, the triple product of covariant vectors is more properly described as a scalar density.
Some authors use "pseudoscalar" to describe an object that looks like a scalar but does not transform like one. Because the triple product transforms as a scalar density not as a scalar, it could be called a "pseudoscalar" by this broader definition. However, the triple product is not a "pseudoscalar density".
When a transformation is an orientation-preserving rotation, its determinant is and the triple product is unchanged. When a transformation is an orientation-reversing rotation then its determinant is and the triple product is negated. An arbitrary transformation could have a determinant that is neither nor .
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 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.
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 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. A proof is provided below.
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 LaplaceâÂÂde Rham operator .
The component of is given by:
Similarly, the and components of are given by:
By combining these three components we obtain:
If geometric algebra is used the cross product b àc of vectors is expressed as their exterior product bâ§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 ÃÂ (b ÃÂ c).
With non-commutative vector operators , special relations hold for the triple product in accordance to
with the unit vectors of , with and being indices of a three dimensional orthonormal basis and the square brackets repressenting the commutator.
Using the individual components of the cross product
as well as the Levi-Civita symbol expression via the Kronecker Delta
we receive the expression
for the first identity and the expression
for the second identity for each of the three indices. By expressing them via a sum of all indices, the original identity is received.
In geometric algebra, three bivectors can also have a triple product. This product mimics the standard triple vector product. The antisymmetric product of three bivectors is.
This proof is made by taking dual of the geometric algebra version of the triple vector product until all vectors become bivectors.
This was three duals. This must also be done to the left side.
By negating both side we obtain:
It can be useful in fields like differential geometry, special relativity and theoretical physics in general to express triple products components using tensor notation.
This is because such a representation provides a basis-invariant (or coordinate-independent) way of expressing the properties of the product.
The triple scalar product is expressed using the Levi-Civita symbol:
while the triple vector product:
referring to the -th component of the resulting vector. This can be simplified by performing a contraction on the Levi-Civita symbols, where is the Kronecker delta function ( when and when ) and is the generalized Kronecker delta function. We can reason out this identity by recognizing that the index will be summed out leaving only and . In the first term, we fix and thus . Likewise, in the second term, we fix and thus .
Returning to the triple cross product,