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==Vector quadruple product == ==Vector quadruple product ==
The '''vector quadruple product''' is defined as the ] of two ]s: The '''vector quadruple product''' is defined as the ] of two cross products:
:<math> \mathbf{a \times b} \mathbf{\times} (\mathbf{c}\times \mathbf{d}) \ ,</math> :<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/> 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> :<math> (\mathbf{a \times b} )\mathbf{\times} (\mathbf{c}\times \mathbf{d}) = \mathbf c - \mathbf d \ ,</math>
using the notation for the ]: 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}\\ :<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}\\
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Equivalent forms can be obtained using the identity: Equivalent forms can be obtained using the identity:
:<math>\mathbf a - \mathbf b -\mathbf d = 0 \ . </math> :<math>\mathbf a - \mathbf b -\mathbf d = 0 \ . </math>


== Interpretation== == Interpretation==

Revision as of 17:26, 16 September 2010

See also: Vector algebra relations

In mathematics, the quadruple product is a product of four vectors in three-dimensional Euclidean space. The name "quadruple product" is used for two different products, the scalar-valued scalar quadruple product and the vector-valued vector quadruple product.

Scalar quadruple product

The scalar quadruple product is defined as the dot product of two cross products:

( a × b ) ⋅ ( c × d )   , {\displaystyle (\mathbf {a\times b} )\mathbf {\cdot } (\mathbf {c} \times \mathbf {d} )\ ,}

where a, b, c, d are vectors in three-dimensional Euclidean space. It can be evaluated using the identity:

( a × b ) ⋅ ( c × d ) = ( a ⋅ c ) ( b ⋅ d ) − ( a ⋅ d ) ( b ⋅ c )   . {\displaystyle (\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} )\ .}

or using the determinant:

( a × b ) ⋅ ( c × d ) = | a ⋅ c a ⋅ d b ⋅ c b ⋅ d |   . {\displaystyle (\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}}\ .}

Vector quadruple product

The vector quadruple product is defined as the cross product of two cross products:

( a × b ) × ( c × d )   , {\displaystyle (\mathbf {a\times b} )\mathbf {\times } (\mathbf {c} \times \mathbf {d} )\ ,}

where a, b, c, d are vectors in three-dimensional Euclidean space. It can be evaluated using the identity:

( a × b ) × ( c × d ) = [ a ,   b ,   d ] c − [ a ,   b ,   c ] d   , {\displaystyle (\mathbf {a\times b} )\mathbf {\times } (\mathbf {c} \times \mathbf {d} )=\mathbf {c} -\mathbf {d} \ ,}

using the notation for the triple product:

[ a ,   b ,   d ] = ( a × b ) ⋅ d = | a ⋅ i ^ a ⋅ j ^ a ⋅ k ^ b ⋅ i ^ b ⋅ j ^ b ⋅ k ^ d ⋅ i ^ d ⋅ j ^ d ⋅ k ^ |   , {\displaystyle =(\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}}\ ,}

where the last form is a determinant with i ^ ,   j ^ ,   k ^ {\displaystyle {\hat {\mathbf {i} }},\ {\hat {\mathbf {j} }},\ {\hat {\mathbf {k} }}} denoting unit vectors along three mutually orthogonal directions.

Equivalent forms can be obtained using the identity:

[ b ,   c ,   d ] a − [ c ,   d ,   a ] b − [ a ,   b ,   c ] d = 0   . {\displaystyle \mathbf {a} -\mathbf {b} -\mathbf {d} =0\ .}

Interpretation

The quadruple products are useful for deriving various formulas in spherical and plane geometry.

References

  1. ^ Edwin Bidwell Wilson, Josiah Willard Gibbs (1901). "§42: Direct and skew products of vectors". Vector analysis: a text-book for the use of students of mathematics. Scribner. pp. 77 ff.

See also

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