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.[2] It can be evaluated using the identity:[2]

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

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.[3] It can be evaluated using the identity:[4]

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

This identity can also be written using tensor notation and the Einstein summation convention as follows:

${\displaystyle (\mathbf {a\times b} )\mathbf {\times } (\mathbf {c} \times \mathbf {d} )=\varepsilon _{ijk}a^{i}c^{j}d^{k}b^{l}-\varepsilon _{ijk}b^{i}c^{j}d^{k}a^{l}=\varepsilon _{ijk}a^{i}b^{j}d^{k}c^{l}-\varepsilon _{ijk}a^{i}b^{j}c^{k}d^{l}}$

using the notation for the triple product:

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

where the last two forms are determinants 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:[5]

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

The quadruple products are useful for deriving various formulas in spherical and plane geometry.[3] For example, if four points are chosen on the unit sphere, A, B, C, D, and unit vectors drawn from the center of the sphere to the four points, a, b, c, d respectively, the identity:

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

in conjunction with the relation for the magnitude of the cross product:

${\displaystyle \|\mathbf {a\times b} \|=ab\sin \theta _{ab}\ ,}$

and the dot product:

${\displaystyle \mathbf {a\cdot b} =ab\cos \theta _{ab}\ ,}$

where a = b = 1 for the unit sphere, results in the identity among the angles attributed to Gauss:

${\displaystyle \sin \theta _{ab}\sin \theta _{cd}\cos x=\cos \theta _{ac}\cos \theta _{bd}-\cos \theta _{ad}\cos \theta _{bc}\ ,}$

where x is the angle between a × b and c × d, or equivalently, between the planes defined by these vectors.

Josiah Willard Gibbs's pioneering work on vector calculus provides several other examples.[3]

• Gibbs, Josiah Willard; Wilson, Edwin Bidwell (1901). Vector analysis: a text-book for the use of students of mathematics. Scribner.

• Binet–Cauchy identity
• Lagrange's identity