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{{see also|Vector algebra relations}} #REDIRECT ]


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In ], the ''']''' is a product of four ] in three-dimensional ]. The name "quadruple product" is used for two different products, the scalar-valued '''scalar quadruple product''' and the vector-valued '''vector quadruple product'''.
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== Scalar quadruple product ==
]
The '''scalar quadruple product''' is defined as the ] of two ]s:
:<math> (\mathbf{a \times b})\mathbf {\cdot}(\mathbf{c}\times \mathbf{d}) \ ,</math>
where '''a, b, c, d''' are vectors in three-dimensional Euclidean space.<ref name=Gibbs/> It can be evaluated using the identity:<ref name=Gibbs>

{{cite book |title=Vector analysis: a text-book for the use of students of mathematics |author=Edwin Bidwell Wilson, Josiah Willard Gibbs |url=http://books.google.com/books?id=RC8PAAAAIAAJ&pg=PA77 |chapter=§42: Direct and skew products of vectors |publisher=Scribner |year=1901 |pages=77 ''ff''}}

</ref>
:<math> (\mathbf{a \times b})\mathbf {\cdot}(\mathbf{c}\times \mathbf{d}) = (\mathbf{a \cdot c})(\mathbf{b \cdot d}) - (\mathbf{a \cdot d})(\mathbf{b \cdot c}) \ . </math>
or using the ]:
:<math>(\mathbf{a \times b})\mathbf {\cdot}(\mathbf{c}\times \mathbf{d}) =\begin{vmatrix} \mathbf{a\cdot c} & \mathbf{a\cdot d} \\
\mathbf{b\cdot c} & \mathbf{b\cdot d} \end{vmatrix} \ . </math>

==Vector quadruple product ==
The '''vector quadruple product''' is defined as the ] of two ]s:
:<math> \mathbf{a \times b} \mathbf{\times} (\mathbf{c}\times \mathbf{d}) \ ,</math>
where '''a, b, c, d''' are vectors in three-dimensional Euclidean space.<ref name=Gibbs/> It can be evaluated using the identity:<ref name=Gibbs/>
:<math> \mathbf{a \times b} \mathbf{\times} (\mathbf{c}\times \mathbf{d}) = \mathbf c - \mathbf d \ ,</math>
using the notation for the ]:
:<math> = (\mathbf{a \times b}) \mathbf{\cdot d } = \begin{vmatrix} \mathbf{a\cdot }\hat {\mathbf i} & \mathbf{a \cdot} \hat {\mathbf j} & \mathbf{a\cdot} \hat {\mathbf k}\\
\mathbf{b\cdot }\hat {\mathbf i} & \mathbf{b\cdot} \hat {\mathbf j} & \mathbf{b\cdot}\hat {\mathbf k}\\ \mathbf{d\cdot} \hat {\mathbf i} & \mathbf{d\cdot} \hat {\mathbf j} & \mathbf{d\cdot }\hat {\mathbf k} \end{vmatrix}\ ,</math>
where the last form is a determinant with <math> \hat {\mathbf i}, \ \hat {\mathbf j}, \ \hat {\mathbf k} </math> denoting unit vectors along three mutually orthogonal directions.

Equivalent forms can be obtained using the identity:
:<math>\mathbf a - \mathbf b -\mathbf d = 0 \ . </math>

== Interpretation==
The quadruple products are useful for deriving various formulas in spherical and plane geometry.<ref name=Gibbs/>

==References==
<references/>

==See also==
*]
*]


]
]
]

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