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-\mathbf{a}\cdot(\mathbf{c}\times \mathbf{b})
</math>
 
== 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 relationships hold:
 
:<math>\mathbf{a}\times (\mathbf{b}\times \mathbf{c}) = (\mathbf{a}\cdot\mathbf{c})\mathbf{b} - (\mathbf{a}\cdot\mathbf{b})\mathbf{c}</math>
:<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>&nbsp;.
 
The first formula 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 ambiguous (see [[Lagrange's formula (disambiguation)|disambiguation page]]). Its right hand member is easier to remember by using the [[mnemonic]] "BAC minus CAB", provided one keeps in mind which vectors are dotted together. A proof is provided below.
 
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> \begin{align}
\nabla \times (\nabla \times \mathbf{f})
& {}= \nabla (\nabla \cdot \mathbf{f} )
- (\nabla \cdot \nabla) \mathbf{f} \\
& {}= \mbox{grad }(\mbox{div } \mathbf{f} )
- \nabla^2 \mathbf{f}.
\end{align} </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>.
 
 
 
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