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Revision as of 17:25, 19 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 ^ b ⋅ i ^ d ⋅ i ^ a ⋅ j ^ b ⋅ j ^ d ⋅ j ^ a ⋅ k ^ b ⋅ k ^ d ⋅ k ^ | = | 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 {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 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 + [ d ,   a ,   b ] c − [ a ,   b ,   c ] d = 0   . {\displaystyle \mathbf {a} -\mathbf {b} +\mathbf {c} -\mathbf {d} =0\ .}

Application

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

( a × b ) ⋅ ( c × d ) = ( a ⋅ c ) ( b ⋅ d ) − ( a ⋅ d ) ( b ⋅ c )   , {\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:

‖ a × b ‖ = a b sin ⁡ θ a b   , {\displaystyle \|\mathbf {a\times b} \|=ab\sin \theta _{ab}\ ,}

and the dot product:

‖ a ⋅ b ‖ = a b cos ⁡ θ a b   , {\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:

sin ⁡ θ a b sin ⁡ θ c d cos ⁡ x = cos ⁡ θ a c cos ⁡ θ b d − cos ⁡ θ a d cos ⁡ θ b c   , {\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.

Gibbs provides several other examples.

Geometric algebra

Main articles: Geometric algebra and Comparison of vector algebra and geometric algebra

Notation

First an orientation is provided to the terminology. There are two different approaches. One is based entirely upon the wedge product, also called the exterior product. To make the connection to cross product clear, an orthonormal set of basis vectors is introduced: e1, e2, e3. The cross product is then:

a × b = | e 1 e 2 e 3 a 1 a 2 a 3 b 1 b 2 b 3 | {\displaystyle \mathbf {a\times b} ={\begin{vmatrix}\mathbf {e_{1}} &\mathbf {e_{2}} &\mathbf {e_{3}} \\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\end{vmatrix}}}

while the wedge product is:

a ∧ b = | e 2 ∧ e 3 e 3 ∧ e 1 e 1 ∧ e 2 a 1 a 2 a 3 b 1 b 2 b 3 | {\displaystyle \mathbf {a\wedge b} ={\begin{vmatrix}\mathbf {e_{2}\wedge e_{3}} &\mathbf {e_{3}\wedge e_{1}} &\mathbf {e_{1}\wedge e_{2}} \\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\end{vmatrix}}}

defined in the bivector space based upon e2∧e3, e3∧e1, e1∧e2. These unit vectors are envisioned as orthogonal oriented unit plane segments in the Euclidean space; for example, e2∧e3 is a unit plane segment with direction e1. Each vector a can be associated with a bivector, called its Hodge dual, or simply dual, denoted *a, by:

∗ a = a 1 e 2 ∧ e 3 + a 2 e 3 ∧ e 1 + a 3 e 1 ∧ e 2 {\displaystyle \mathbf {*a} =a_{1}\mathbf {e_{2}\wedge e_{3}} +a_{2}\mathbf {e_{3}\wedge e_{1}} +a_{3}\mathbf {e_{1}\wedge e_{2}} }

resulting in the relations:

a ∧ b = ∗ ( a × b )   , {\displaystyle \mathbf {a\wedge b} =*(\mathbf {a\times b} )\ ,}
a × b = ∗ ( a ∧ b ) . {\displaystyle \mathbf {a\times b} =*(\mathbf {a\wedge b} ).}

A comparison of the cross product and the bivector can be found in this article.

Using these same basis vectors, the exterior algebra (also known as the Grassmann algebra) for this space is based upon:

1 {\displaystyle 1}                   the scalar
e 1 ,   e 2 ,   e 3 {\displaystyle \mathbf {e_{1},\ e_{2},\ e_{3}} }             vectors
e 1 ∧ e 2 ,   e 2 ∧ e 3 ,   e 3 ∧ e 1 {\displaystyle \mathbf {e_{1}\wedge e_{2},\ e_{2}\wedge e_{3},\ e_{3}\wedge e_{1}} }    the bivectors
I = e 1 ∧ e 2 ∧ e 3 {\displaystyle {\mathit {I}}=\mathbf {e_{1}\wedge e_{2}\wedge e_{3}} }           oriented unit volume element

A second approach is based upon the Clifford product or geometric product. The geometric product of two vectors a and b (denoted ab) includes both the dot (scalar) product and the wedge product:

a b = a ⋅ b + a ∧ b   . {\displaystyle \mathbf {ab} =\mathbf {a\cdot b} +\mathbf {a\wedge b} \ .}

The dot product is the same as the dot product already known from vector algebra. The Clifford algebra for this space is based upon:

1 {\displaystyle 1}                    the scalar
e 1 ,   e 2 ,   e 3 {\displaystyle \mathbf {e_{1},\ e_{2},\ e_{3}} }             vectors
e 1 e 2 ,   e 2 e 3 ,   e 3 e 1 {\displaystyle \mathbf {e_{1}e_{2},\ e_{2}e_{3},\ e_{3}e_{1}} }        the bivectors; also denoted by e 12 ,   e 23 ,   e 31 {\displaystyle \mathbf {e_{12},\ e_{23},\ e_{31}} }
I = e 1 e 2 e 3 {\displaystyle {\mathit {I}}=\mathbf {e_{1}e_{2}e_{3}} }            oriented unit volume element; also denoted by e 123 {\displaystyle \mathbf {e_{123}} }

Notice that, because of orthogonality:

e 1 e 2 = ( e 1 ⋅ e 2 ) + e 1 ∧ e 2 = e 1 ∧ e 2   , {\displaystyle \mathbf {e_{1}e_{2}} =(\mathbf {e_{1}\cdot e_{2}} )+\mathbf {e_{1}\wedge e_{2}} =\mathbf {e_{1}\wedge e_{2}} \ ,}

so the bivectors are the same for both algebras. Also,

e 1 e 2 e 3 = ( e 1 ⋅ e 2 + e 1 ∧ e 2 ) e 3 = ( e 1 ∧ e 2 ) e 3 = ( e 1 ∧ e 2 ) ⋅ e 3 + ( e 1 ∧ e 2 ) ∧ e 3 {\displaystyle \mathbf {e_{1}e_{2}e_{3}} =\left(\mathbf {e_{1}\cdot e_{2}+e_{1}\wedge e_{2}} \right)\mathbf {e_{3}} =\left(\mathbf {e_{1}\wedge e_{2}} \right)\mathbf {e_{3}} =\left(\mathbf {e_{1}\wedge e_{2}} \right)\mathbf {\cdot e_{3}} +\left(\mathbf {e_{1}\wedge e_{2}} \right)\mathbf {\wedge e_{3}} }

Now the scalar triple product is given by:

( a ∧ b ) ⋅ c = ( a ⋅ c ) b − ( b ⋅ c ) a   , {\displaystyle (\mathbf {a\wedge b} )\mathbf {\cdot c} =(\mathbf {a\cdot c} )\mathbf {b} -(\mathbf {b\cdot c} )\mathbf {a} \ ,}

so, by orthogonality of the basis vectors, the oriented unit volumes are the same in both algebras.

I = e 1 e 2 e 3 = e 1 ∧ e 2 ∧ e 3   . {\displaystyle {\mathit {I}}=\mathbf {e_{1}e_{2}e_{3}} =\mathbf {e_{1}\wedge e_{2}\wedge e_{3}} \ .}

In the Clifford algebra, the cross product and the wedge product are then related by:

a × b = − a ∧ b   e 123 {\displaystyle \mathbf {a\times b} =-\mathbf {a\wedge b\ e_{123}} }

Thus, the wedge product in three dimensions is related to the cross product from vector algebra in two different ways that differ only in sign in some spaces.

Various products of four vectors

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This section explores the quadruple products of vectors using only the approach of geometric algebra. In geometric algebra there are many possible quadruple products. For example, abcd, (a·b)cd, (a∧b)·(c∧d), (a∧b)(c∧d), (a∧b)∧(c∧d) and so forth. Some of these are evaluated in simpler terms below.

The geometric product of two bivectors is:

( a ∧ b ) ( c ∧ d ) = ( a b − a ⋅ b ) ( c d − c ⋅ d ) = a b c d − ( a b ) ( c ⋅ d ) − ( c d ) ( a ⋅ b ) + ( a ⋅ b ) ( c ⋅ d ) {\displaystyle (\mathbf {a\wedge b} )(\mathbf {c\wedge d} )=(\mathbf {ab-a\cdot b} )(\mathbf {cd-c\cdot d} )=\mathbf {abcd} -(\mathbf {ab} )(\mathbf {c\cdot d} )-(\mathbf {cd} )(\mathbf {a\cdot b} )+(\mathbf {a\cdot b} )(\mathbf {c\cdot d} )}

Consequently, if a·b and c·d are zero:

( a ∧ b ) ( c ∧ d ) = a b c d {\displaystyle (\mathbf {a\wedge b} )(\mathbf {c\wedge d} )=\mathbf {abcd} }

In particular, for orthonormal basis vectors, the product of two bivectors is another basis bivector:

( e 1 ∧ e 2 ) ( e 2 ∧ e 3 ) = ( e 1 e 2 ) ( e 2 e 3 ) = e 1 ( e 2 e 2 ) e 3 = e 1 e 3 = e 1 ∧ e 3 {\displaystyle (\mathbf {e} _{1}\wedge \mathbf {e} _{2})(\mathbf {e} _{2}\wedge \mathbf {e} _{3})=(\mathbf {e_{1}e_{2}} )(\mathbf {e_{2}e_{3}} )=\mathbf {e_{1}} (\mathbf {e_{2}e_{2}} )\mathbf {e_{3}} =\mathbf {e_{1}e_{3}} =\mathbf {e_{1}\wedge e_{3}} }

Notes

  1. Gibbs & Wilson 1901, §42 of section "Direct and skew products of vectors", p.77
  2. Gibbs & Wilson 1901, p. 76
  3. Gibbs & Wilson 1901, p. 76
  4. Gibbs & Wilson 1901, pp. 77 ff
  5. Gibbs & Wilson 1901, p. 77
  6. Gibbs & Wilson, Equation 27, p. 77 harvnb error: no target: CITEREFGibbsWilson (help)
  7. Gibbs & Wilson 1901, pp. 77 ff
  8. Gibbs & Wilson 1901, pp. 77 ff
  9. Lounesto 2001, p. 37
  10. Lounesto 2001, p. 36
  11. Lounesto 2001, p. 39
  12. Lounesto 2001, p. 53
  13. Lounesto 2001, p. 41
  14. Doran & Lasenby 2003, p. 32
  15. Lounesto 2001, p. 95
  16. Hertrich-Jeromin 2003, §6.4: Spheres and the Clifford dual, p. 289
  17. Doran & Lasenby 2003, p. 33

References

See also

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