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{{about|ternary operations on vectors|the identity in number theory|Jacobi triple product|the product in nuclear fusion|Lawson criterion}}

In [[vector calculus]], a branch of [[mathematics]], the '''triple product''' is a product of three 3-[[dimension (vector space)|dimensional]] vectors, usually [[Euclidean vector]]s. 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 ==
[[Image:Parallelepiped volume.svg|right|thumb|240px|Three vectors defining a parallelepiped]]

The '''scalar triple product''' (also called the '''mixed''' or '''box 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
:<math> \mathbf{a}\cdot(\mathbf{b}\times \mathbf{c}) </math>
is the (signed) [[volume]] of the [[parallelepiped]] defined by the three vectors given.

=== Properties ===
* The scalar triple product is invariant under a [[circular shift]] of its three operands ('''a''', '''b''', '''c'''):
::<math>
\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c})=
\mathbf{b}\cdot(\mathbf{c}\times \mathbf{a})=
\mathbf{c}\cdot(\mathbf{a}\times \mathbf{b})
</math>

* 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.
::<math>
\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c}) =
(\mathbf{a}\times \mathbf{b})\cdot \mathbf{c}
</math>

* Switching the two vectors in the cross product [[additive inverse|negates]] the triple product:
::<math>
\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c}) =
-\mathbf{a}\cdot(\mathbf{c}\times \mathbf{b})
</math>
: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.

* The scalar triple product can also be understood as the [[determinant]] of the {{gaps|3|×|3}} matrix having the three vectors either as its rows or its columns (a matrix has the same determinant as its [[transpose]]):
::<math>\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c}) = \det \begin{bmatrix}
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3 \\
\end{bmatrix}.</math>

* If the scalar triple product is equal to zero, then the three vectors '''a''', '''b''', and '''c''' are [[coplanar]], since the "parallelepiped" defined by them would be flat and have no volume.

* If any two vectors of triple scalar product are equal, then its value is zero:
::<math>
\mathbf{a} \cdot (\mathbf{a} \times \mathbf{b}) =
\mathbf{a} \cdot (\mathbf{b} \times \mathbf{a}) =
\mathbf{a} \cdot (\mathbf{b} \times \mathbf{b}) =
\mathbf{a} \cdot (\mathbf{a} \times \mathbf{a}) = 0
</math>

* Moreover,
::<math>
[\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c})] \mathbf{a} =
(\mathbf{a}\times \mathbf{b})\times (\mathbf{a}\times \mathbf{c})
</math>

* The [[multiplication|simple product]] of two triple products (or the square of a triple product), may be expanded in terms of dot products
::<math>((\mathbf{a}\times \mathbf{b})\cdot \mathbf{c})\;((\mathbf{d}\times \mathbf{e})\cdot \mathbf{f}) = \det\left[ \begin{pmatrix}
\mathbf{a} \\
\mathbf{b} \\
\mathbf{c}
\end{pmatrix}\cdot \begin{pmatrix}
\mathbf{d} & \mathbf{e} & \mathbf{f}
\end{pmatrix}\right] = \det \begin{bmatrix}
\mathbf{a}\cdot \mathbf{d} & \mathbf{a}\cdot \mathbf{e} & \mathbf{a}\cdot \mathbf{f} \\
\mathbf{b}\cdot \mathbf{d} & \mathbf{b}\cdot \mathbf{e} & \mathbf{b}\cdot \mathbf{f} \\
\mathbf{c}\cdot \mathbf{d} & \mathbf{c}\cdot \mathbf{e} & \mathbf{c}\cdot \mathbf{f}
\end{bmatrix}</math><ref name="Wong">{{cite book
|title=Introduction to Mathematical Physics: Methods & Concepts
|first=Chun Wa
|last=Wong
|publisher=Oxford University Press
|isbn=9780199641390
|year=2013
|pages=215
|url=http://books.google.co.uk/books?id=JePI32FCqBYC&pg=PA215#v=onepage&f=true
}}</ref>

===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 (vector space)|orientation]] of the frame or the [[parity of a permutation|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 [[cross product#Cross product and handedness|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 (mathematics)|rotation operator]], then
:<math>
\mathbf{Ta} \cdot (\mathbf{Tb} \times \mathbf{Tc}) =
\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}),
</math>
but if '''T''' is an [[improper rotation]], then
:<math>
\mathbf{Ta} \cdot (\mathbf{Tb} \times \mathbf{Tc}) =
-\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}).
</math>

===As an exterior product===
[[Image:Exterior calc triple product.svg|thumb|right|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
:<math>\mathbf{a} \wedge \mathbf{b} \wedge \mathbf{c}</math>
is a trivector with magnitude equal to the scalar triple product, and is the [[Hodge dual]] of the triple product. As the exterior product is associative brackets are not needed as it does not matter which of {{nowrap|'''a''' ∧ '''b'''}} or {{nowrap|'''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 {{nowrap|'''a''' ∧ '''b'''}}, {{nowrap|'''b''' ∧ '''c'''}} and {{nowrap|'''a''' ∧ '''c'''}} 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 [[tensor contraction|contraction]] of vectors with a rank-3 tensor equivalent to the form (or a [[pseudotensor]] equivalent to the volume pseudoform); see [[#Interpretations|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:

:<math>\mathbf{a}\times (\mathbf{b}\times \mathbf{c}) = \mathbf{b}(\mathbf{a}\cdot\mathbf{c}) - \mathbf{c}(\mathbf{a}\cdot\mathbf{b})</math>.

This is known as '''triple product expansion''', or '''Lagrange's formula''',<ref>[[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: see {{cite book|author=Lagrange, J-L|title=Oeuvres|volume=vol 3|chapter=Solutions analytiques de quelques problèmes sur les pyramides triangulaires|year=1773}} He may have written a formula similar to the triple product expansion in component form. See also [[Lagrange's identity]] and {{cite book|author=[[Kiyoshi Itō]]|title=Encyclopedic Dictionary of Mathematics|year=1987|isbn=0-262-59020-4|publisher=MIT Press|page=1679}}</ref><ref name=Itô>
{{cite book |title=Encyclopedic dictionary of mathematics |author=[[Kiyoshi Itō]] |page=1679 |chapter=§C: Vector product |url=http://books.google.com/books?id=azS2ktxrz3EC&pg=PA1679 |isbn=0-262-59020-4 |edition=2nd |publisher=MIT Press |year=1993}}

</ref>
although the latter name is also used for [[Lagrange's formula (disambiguation)|several other formulae]]. Its right hand side can be remembered by using the [[mnemonic]] "BAC − CAB", provided one keeps in mind which vectors are dotted together. A proof is provided [[#Tensor calculus|below]].

Since the cross product is anticommutative, this formula may also be written (up to permutation of the letters) as:

:<math>(\mathbf{a}\times \mathbf{b})\times \mathbf{c} = -\mathbf{c}\times(\mathbf{a}\times \mathbf{b}) = -(\mathbf{c}\cdot\mathbf{b})\mathbf{a} + (\mathbf{c}\cdot\mathbf{a})\mathbf{b}</math>

From Lagrange's formula it follows that the vector triple product satisfies:

:<math>\mathbf{a}\times (\mathbf{b}\times \mathbf{c}) \; + \mathbf{b}\times (\mathbf{c}\times \mathbf{a}) \; + \mathbf{c}\times (\mathbf{a}\times \mathbf{b}) = 0</math>

which is the [[Jacobi identity]] for the cross product

These formulas are very useful in simplifying vector calculations in [[physics]]. A related identity regarding [[gradient]]s and useful in [[vector calculus]] is Lagrange's formula of vector cross-product identity:<ref name= Lin>

{{cite book |title=Numerical Modelling of Water Waves: An Introduction to Engineers and Scientists |author=Pengzhi Lin |page=13 |url=http://books.google.com/books?id=x6ALwaliu5YC&pg=PA13 |isbn=0-415-41578-0 |year=2008 |publisher=Routledge}}

</ref>
:<math>
\boldsymbol{\nabla} \times (\boldsymbol{\nabla} \times \mathbf{f}) = \boldsymbol{\nabla} (\boldsymbol{\nabla} \cdot \mathbf{f}) - (\boldsymbol{\nabla} \cdot \boldsymbol{\nabla}) \mathbf{f}
</math>
This can be also regarded as a special case of the more general [[Laplace–Beltrami operator|Laplace–de Rham operator]] <math>\Delta = d \delta + \delta d</math>.

===Proof===
The <math>x</math> component of <math>\mathbf{u}\times (\mathbf{v}\times \mathbf{w})</math> is given by:

:<math>\mathbf{u}_y(\mathbf{v}_x\mathbf{w}_y-\mathbf{v}_y\mathbf{w}_x)-\mathbf{u}_z(\mathbf{v}_z\mathbf{w}_x-\mathbf{v}_x\mathbf{w}_z)</math>

or

:<math>\mathbf{v}_x(\mathbf{u}_y\mathbf{w}_y+\mathbf{u}_z\mathbf{w}_z)-\mathbf{w}_x(\mathbf{u}_y\mathbf{v}_y+\mathbf{u}_z\mathbf{v}_z)</math>

By adding and subtracting <math>\mathbf{u}_x\mathbf{v}_x\mathbf{w}_x</math>, this becomes

:<math>\mathbf{v}_x(\mathbf{u}_x\mathbf{w}_x+\mathbf{u}_y\mathbf{w}_y+\mathbf{u}_z\mathbf{w}_z)-\mathbf{w}_x(\mathbf{u}_x\mathbf{v}_x+\mathbf{u}_y\mathbf{v}_y+\mathbf{u}_z\mathbf{v}_z)=(\mathbf{u}\cdot\mathbf{w})\mathbf{v}_x-(\mathbf{u}\cdot\mathbf{v})\mathbf{w}_x</math>

Similarly, the <math>y</math> and <math>z</math> components of <math>\mathbf{u}\times (\mathbf{v}\times \mathbf{w})</math> are given by:

:<math> (\mathbf{u}\cdot\mathbf{w})\mathbf{v}_y-(\mathbf{u}\cdot\mathbf{v})\mathbf{w}_y</math>

and

:<math> (\mathbf{u}\cdot\mathbf{w})\mathbf{v}_z-(\mathbf{u}\cdot\mathbf{v})\mathbf{w}_z</math>

By combining these three components we obtain:

:<math>\mathbf{u}\times (\mathbf{v}\times \mathbf{w}) = (\mathbf{u}\cdot\mathbf{w})\ \mathbf{v} - (\mathbf{u}\cdot\mathbf{v})\ \mathbf{w}</math><ref>{{cite book|title=Mathematical Methods in Science and Engineering|author=J. Heading|publisher=American Elsevier Publishing Company, Inc|pages=262–263|year=1970}}</ref>

===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 [[geometric algebra #Extensions of the inner and outer products|left contraction]]<ref name=Lounesto>
{{cite book |author=Pertti Lounesto |page=46 |title=Clifford algebras and spinors |isbn=0-521-00551-5 |edition=2nd |publisher=Cambridge University Press |year=2001}}</ref> can be used, so the formula becomes<ref name=Personen>{{cite web|title= Geometric Algebra of One and Many Multivector Variables|Author=Janne Personen|url=http://www.helsinki.fi/%7Ejmpesone/index_files/GA_files/Chapter_1.pdf|page=37}}</ref>

<math>\begin{align}
-\mathbf{a} \;\big\lrcorner\; (\mathbf{b} \wedge \mathbf{c}) &= \mathbf{b} \wedge (\mathbf{a} \;\big\lrcorner\; \mathbf{c}) - (\mathbf{a} \;\big\lrcorner\; \mathbf{b}) \wedge \mathbf{c} \\
&= (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \end{align}</math>

The proof follows from the properties of the contraction.<ref name=Lounesto/> The result is the same vector as calculated using '''a''' × ('''b''' × '''c''').

== Interpretations ==
{{unreferenced|section|date=January 2014}}
=== Tensor calculus ===
In [[tensor calculus|tensor notation]] the triple product is expressed using the [[Levi-Civita symbol]]:
: <math>(\mathbf{a} \cdot (\mathbf{b}\times \mathbf{c})) = \varepsilon_{ijk} a^i b^j c^k</math>
and
: <math>(\mathbf{a} \times (\mathbf{b}\times \mathbf{c}))_i = \varepsilon_{ijk} a^j \varepsilon_{k\ell m} b^\ell c^m = \varepsilon_{ijk}\varepsilon_{k\ell m} a^j b^\ell c^m</math>

which can be simplified by performing a [[tensor contraction|contraction]] on the Levi-Civita symbols, <math>\varepsilon_{ijk} \varepsilon_{k\ell m}=\delta_{i\ell}\delta_{jm}-\delta_{im}\delta_{\ell j}</math> and simplifying the result.
{{Expand section|date=January 2014}}

== Notes ==
<references/>

==References==
*{{cite book|last = Lass|first = Harry|title = Vector and Tensor Analysis|publisher = McGraw-Hill Book Company, Inc.|year = 1950|pages = 23–25}}

[[Category:Vector calculus]]
[[Category:Ternary operations]]
[[Category:Vectors]]

Truncated power function - Revision history

en>Helpful Pixie Bot: ISBNs (Build KH)