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,[1] scalar quadruple product and the vector-valued vector quadruple product or vector product of four vectors .
Scalar quadruple product [ ] The scalar quadruple product is defined as the dot product of two cross products :
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{\displaystyle (\mathbf {a\times b} )\cdot (\mathbf {c} \times \mathbf {d} )\ ,}
where a, b, c, d are vectors in three-dimensional Euclidean space.[2] [2]
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{\displaystyle (\mathbf {a\times b} )\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 :
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{\displaystyle (\mathbf {a\times b} )\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:
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{\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] [4]
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{\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:
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{\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 :
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{\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
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{\displaystyle {\hat {\mathbf {i} }},\ {\hat {\mathbf {j} }},\ {\hat {\mathbf {k} }}}
Equivalent forms can be obtained using the identity:[5]
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{\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\ .}
Application [ ] The quadruple products are useful for deriving various formulas in spherical and plane geometry.[3] 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:
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{\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:
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{\displaystyle \|\mathbf {a\times b} \|=ab\sin \theta _{ab}\ ,}
and the dot product:
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{\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:
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{\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]
Notes [ ] References [ ] See also [ ] 
![({\mathbf {a\times b}}){\mathbf {\times }}({\mathbf {c}}\times {\mathbf {d}})=[{\mathbf {a,\ b,\ d}}]{\mathbf c}-[{\mathbf {a,\ b,\ c}}]{\mathbf d}\ ,](https://wikimedia.org/api/rest_v1/media/math/render/svg/83e41e3db9c5e3a2fb2030f9822ab0af1f24ad91)
![[{\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}}\ ,](https://wikimedia.org/api/rest_v1/media/math/render/svg/5130be70093245e5bc521cbb5a7429259594fd9d)
![[{\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\ .](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aab120a97d92e8ffd4c390fc8be37ebdf564021)
