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Ingeometry andalgebra, thetriple product is a product of three 3-dimensional vectors, usuallyEuclidean vectors. The name "triple product" is used for two different products, thescalar-valuedscalar triple product and, less often, thevector-valuedvector triple product.
Thescalar triple product (also called themixed product,box product, ortriple scalar product) is defined as thedot product of one of the vectors with thecross product of the other two.
The scalar triple product is unchanged under acircular shift of its three operands (a,b,c):
Swapping the positions of the operators without re-ordering the operands leaves the triple product unchanged. This follows from the preceding property and the commutative property of the dot product:
Swapping any two of the three operandsnegates the triple product. This follows from the circular-shift property and theanticommutativity of the cross product:
The scalar triple product can also be understood as thedeterminant of the3×3 matrix that has the three vectors either as its rows or its columns (a matrix has the same determinant as itstranspose):
If the scalar triple product is equal to zero, then the three vectorsa,b, andc arecoplanar, since the parallelepiped defined by them would be flat and have no volume.
If any two vectors in the scalar triple product are equal, then its value is zero:
Also:
Thesimple product of two triple products (or the square of a triple product), may be expanded in terms of dot products:[1]This restates in vector notation that the product of the determinants of two3 × 3 matrices equals the determinant of their matrix product. As a special case, the square of a triple product is aGram determinant. Note that this determinant is well defined for vectors inRm (m-dimensionalEuclidean space) even whenm ≠ 3; in particular, theabsolute value of a triple product for three vectors inRm can be computed from this formula for the square of a triple product by taking its square root:
The ratio of the triple product and the product of the three vector norms is known as apolar sine:which ranges between −1 and 1.
Strictly speaking, ascalar 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 ascalar 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+1 and the triple product is unchanged. When a transformation is an orientation-reversing rotation then its determinant is−1 and the triple product is negated. An arbitrary transformation could have a determinant that is neither+1 nor−1.
The three vectors spanning a parallelepiped have triple product equal to its volume. (However, beware that the direction of the arrows in this diagram is incorrect.)
Inexterior algebra andgeometric algebra the exterior product of two vectors is abivector, while the exterior product of three vectors is atrivector. 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 vectorsa,b andc, the product
is a trivector with magnitude equal to the scalar triple product, i.e.
,
and is theHodge dual of the scalar triple product. As the exterior product is associative brackets are not needed as it does not matter which ofa ∧b orb ∧c is calculated first, though the order of the vectors in the product does matter. Geometrically the trivectora ∧b ∧c corresponds to the parallelepiped spanned bya,b, andc, with bivectorsa ∧b,b ∧c anda ∧c matching theparallelogram faces of the parallelepiped.
The triple product is identical to thevolume form of the Euclidean 3-space applied to the vectors viainterior product. It also can be expressed as acontraction of vectors with a rank-3 tensor equivalent to the form (or apseudotensor equivalent to the volume pseudoform); seebelow.
Thevector triple product is defined as thecross product of one vector with the cross product of the other two. The following relationship holds:
.
This is known astriple product expansion, orLagrange's formula,[2][3] although the latter name is also used forseveral other formulas. Its right hand side can be remembered by using themnemonic "ACB − ABC", provided one keeps in mind which vectors are dotted together. A proof is providedbelow. Some textbooks write the identity as such that a more familiarmnemonic "BAC − CAB" is obtained, 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 theJacobi identity for the cross product. Another useful formula follows:
These formulas are very useful in simplifying vector calculations inphysics. A related identity regardinggradients and useful invector calculus is Lagrange's formula of vector cross-product identity:[4]
If geometric algebra is used the cross productb ×c of vectors is expressed as their exterior productb∧c, abivector. The second cross product cannot be expressed as an exterior product, otherwise the scalar triple product would result. Instead aleft contraction[6] can be used, so the formula becomes[7]
The proof follows from the properties of the contraction.[6] The result is the same vector as calculated usinga × (b ×c).
In geometric algebra, threebivectors can also have a triple product. This product mimics the standard triple vector product. The antisymmetric product of three bivectors is.
This is because such a representation provides abasis-invariant (orcoordinate-independent) way of expressing the properties of the product.
The triple scalar product is expressed using theLevi-Civita symbol:[8]while the triple vector product:referring to the-th component of the resulting vector. This can be simplified by performing acontraction on theLevi-Civita symbols,where is theKronecker delta function ( when and when) and is thegeneralized 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.
^Joseph Louis Lagrange did not develop the cross product as an algebraic product on vectors, but did use an equivalent form of it in components: seeLagrange, J-L (1773). "Solutions analytiques de quelques problèmes sur les pyramides triangulaires".Oeuvres. Vol. 3. He may have written a formula similar to the triple product expansion in component form. See alsoLagrange's identity andKiyosi Itô (1987).Encyclopedic Dictionary of Mathematics. MIT Press. p. 1679.ISBN0-262-59020-4.
