Quadruple product

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.

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

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

where a, b, c, d are vectors in three-dimensional Euclidean space.^[1] It can be evaluated using the identity:^[1]

${\displaystyle (\mathbf {a\times b} )\mathbf {\cdot } (\mathbf {c} \times \mathbf {d} )=(\mathbf {a\cdot c} )(\mathbf {b\cdot d} )-(\mathbf {a\cdot d} )(\mathbf {b\cdot c} )\ .}$

or using the determinant:

${\displaystyle (\mathbf {a\times b} )\mathbf {\cdot } (\mathbf {c} \times \mathbf {d} )={\begin{vmatrix}\mathbf {a\cdot c} &\mathbf {a\cdot d} \\\mathbf {b\cdot c} &\mathbf {b\cdot d} \end{vmatrix}}\ .}$

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

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

where a, b, c, d are vectors in three-dimensional Euclidean space.^[1] It can be evaluated using the identity:^[1]

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

using the notation for the triple product:

${\displaystyle [\mathbf {a,\ b,\ d} ]=(\mathbf {a\times b} )\mathbf {\cdot d} ={\begin{vmatrix}\mathbf {a\cdot } {\hat {\mathbf {i} }}&\mathbf {a\cdot } {\hat {\mathbf {j} }}&\mathbf {a\cdot } {\hat {\mathbf {k} }}\\\mathbf {b\cdot } {\hat {\mathbf {i} }}&\mathbf {b\cdot } {\hat {\mathbf {j} }}&\mathbf {b\cdot } {\hat {\mathbf {k} }}\\\mathbf {d\cdot } {\hat {\mathbf {i} }}&\mathbf {d\cdot } {\hat {\mathbf {j} }}&\mathbf {d\cdot } {\hat {\mathbf {k} }}\end{vmatrix}}\ ,}$

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

Equivalent forms can be obtained using the identity:

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

The quadruple products are useful for deriving various formulas in spherical and plane geometry.^[1]