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Quadruple product

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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.
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