mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the quadruple product is a product of four vectors in three-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

. The name "quadruple product" is used for two different products, the scalar-valued 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 In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...

of two

cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...

s: :

(\mathbf)\cdot(\mathbf\times \mathbf) \ ,

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

(\mathbf)\cdot(\mathbf\times \mathbf) = (\mathbf)(\mathbf) - (\mathbf)(\mathbf) \ .

or using the

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...

: :

(\mathbf)\cdot(\mathbf\times \mathbf) =\begin \mathbf & \mathbf \\
 \mathbf & \mathbf  \end \ .

Proof

We first prove that :

\begin
  \mathbf \times (\mathbf \times \mathbf) \cdot \mathbf = (\mathbf \times \mathbf) \cdot (\mathbf \times \mathbf).
\end

This can be shown by straightforward matrix algebra using the correspondence between elements of

\mathbb^3

and

\mathfrak(3)

, given by

\mathbb^3 \ni \mathbf = \begin a_1 & a_2 & a_3 \end^\mathrm \mapsto \mathbf \in \mathfrak(3)

, where :

\begin
  \mathbf = \begin
    0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0
  \end.
\end

It then follows from the properties of

skew-symmetric matrices In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if a_ ...

that :

\begin
  \mathbf \times (\mathbf \times \mathbf) \cdot \mathbf = (\mathbf\mathbf\mathbf)^\mathrm \mathbf = \mathbf^\mathrm \mathbf \mathbf\mathbf = (-\mathbf\mathbf)^\mathrm \mathbf\mathbf = (\mathbf\mathbf)^\mathrm\mathbf\mathbf = (\mathbf \times \mathbf) \cdot (\mathbf \times \mathbf).
\end

We also know from vector triple products that :

\begin
  \mathbf \times (\mathbf \times \mathbf) = (\mathbf \cdot \mathbf)\mathbf - (\mathbf \cdot \mathbf)\mathbf.
\end

Using this identity along with the one we have just derived, we obtain the desired identity: :

\cdot \mathbf = (\mathbf \cdot \mathbf)(\mathbf \cdot \mathbf) - (\mathbf \cdot \mathbf)(\mathbf \cdot \mathbf). \end

Vector quadruple product

The vector quadruple product is defined as the

of two cross products: :

(\mathbf) \mathbf (\mathbf\times \mathbf) \ ,

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

\mathbf d \ ,

using the notation for the

triple product In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector- ...

: :

= \mathbf \cdot (\mathbf \times \mathbf)\ .

Equivalent forms can be obtained using the identity: :

mathbf d = 0 \ .

This identity can also be written using

tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...

notation and the

Einstein summation In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...

convention as follows: :

(\mathbf )\mathbf (\mathbf\times \mathbf)=\varepsilon_ a^i c^j d^k b^l - \varepsilon_ b^i c^j d^k a^l=\varepsilon_ a^i b^j d^k c^l - \varepsilon_ a^i b^j c^k d^l

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: :

(\mathbf)\mathbf(\mathbf) = (\mathbf )(\mathbf )-(\mathbf)(\mathbf ) \ ,

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

\, \mathbf\,  = a b \sin \theta_ \ ,

and the dot product: :

\mathbf = a b \cos \theta_ \ ,

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

\sin \theta_\sin \theta_\cos x = \cos\theta_\cos\theta_ - \cos\theta_ \cos \theta_ \ ,

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

Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...

's pioneering work on vector calculus provides several other examples.

Notes

References