Seven-dimensional cross product

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In mathematics, the seven-dimensional cross product is a binary operation on vectors in a seven-dimensional Euclidean space. It is a generalization of the ordinary three-dimensional cross product. The seven-dimensional cross product has the same relationship to the octonions as the three-dimensional cross product does to the quaternions.

Nontrivial binary cross products exist only in 3 and 7 dimensions. There are no higher-dimensional analogs.

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[edit] Characteristic properties

A cross product in an n-dimensional Euclidean space V is defined as a bilinear map

V × V → V

such that

  • x · (x × y) = y · (x × y) = 0, and
  • |x × y|2 = |x|2 |y|2 − (x · y)2

for all x and y in V. Here x·y denotes the standard Euclidean dot product. The first of these properties states that the cross product should be perpendicular to each of its arguments. The second states that the norm of the cross product should be equal to the area of the parallelogram formed by the arguments. This is equivalent to the statement

  • |x × y| = |x||y|sin θ

where θ is the inner angle between x and y.

It has been shown that nontrivial cross products exist only for n = 3 and n = 7. The reason has to do with the connection between cross products and normed division algebras (see the section below).

Some properties that follow from the above characterization, and are therefore true in seven dimensions, are:

  • x × y = −y × x
  • x · (y × z) = y · (z × x) = z · (x × y)
  • x × (x × y) = −|x|2y + (x·y)x.

Some properties of the 3-dimensional cross product which do not generally hold for the 7-dimensional one are the vector triple product formula (equality holds in three dimensions)

x × (y × z) ≠ (x·z)y − (x·y)z

and the Jacobi identity:

x × (y × z) + y × (z × x) + z × (x × y) ≠ 0.

Both of these fail due to the nonassociativity of the octonions.

[edit] Coordinate expression

Unlike the 3-dimensional cross product, the 7-dimension cross product is not unique (up to a sign). This is because there is more than one direction perpendicular to any given plane. However, any two cross products differ by an orthogonal transformation.

One possible cross product on R7 is given by the rules

  • e1×e2 = e4
  • ei×ej = ek implies that
    • e2i×e2j = e2k
    • ei+1×ej+1 = ek+1

where {ei} is the standard basis for R7 and the indices are read from 1 to 7 modulo 7. Together with the fact that the cross product must be antisymmetric these rules complete determine the cross product.

Explicitly, the cross product is given by the expression

\begin{align}\mathbf{x}\times\mathbf{y} = (x_2y_4 - x_4y_2 + x_3y_7 - x_7y_3 + x_5y_6 - x_6y_5)\,&\mathbf{e}_1 \\ {}+ (x_3y_5 - x_5y_3 + x_4y_1 - x_1y_4 + x_6y_7 - x_7y_6)\,&\mathbf{e}_2 \\ {}+ (x_4y_6 - x_6y_4 + x_5y_2 - x_2y_5 + x_7y_1 - x_1y_7)\,&\mathbf{e}_3 \\ {}+ (x_5y_7 - x_7y_5 + x_6y_3 - x_3y_6 + x_1y_2 - x_2y_1)\,&\mathbf{e}_4 \\ {}+ (x_6y_1 - x_1y_6 + x_7y_4 - x_4y_7 + x_2y_3 - x_3y_2)\,&\mathbf{e}_5 \\ {}+ (x_7y_2 - x_2y_7 + x_1y_5 - x_5y_1 + x_3y_4 - x_4y_3)\,&\mathbf{e}_6 \\ {}+ (x_1y_3 - x_3y_1 + x_2y_6 - x_6y_2 + x_4y_5 - x_5y_4)\,&\mathbf{e}_7. \\ \end{align}

The 7-dimensional cross product can also be written as a sum of seven 3-dimensional cross products. Let πi denote the orthogonal projection of R7 onto the 3-dimensional subspace spanned by ei+1, ei+2, and ei+4. Then

\mathbf{x}\times\mathbf{y} = \sum_{i=1}^{7}\pi_i(\mathbf x)\times\pi_i(\mathbf y).

Since the cross product is bilinear, one can write the operator x×– as a matrix. This matrix has the form

T_{\mathbf x} = \left[\begin{matrix} 0 & -x_4 & -x_7 & x_2 & -x_6 & x_5 & x_3 \\ x_4 & 0 & -x_5 & -x_1 & x_3 & -x_7 & x_6 \\ x_7 & x_5 & 0 & -x_6 & -x_2 & x_4 & -x_1 \\ -x_2 & x_1 & x_6 & 0 & -x_7 & -x_3 & x_5 \\ x_6 & -x_3 & x_2 & x_7 & 0 & -x_1 & -x_4 \\ -x_5 & x_7 & -x_4 & x_3 & x_1 & 0 & -x_2 \\ -x_3 & -x_6 & x_1 & -x_5 & x_4 & x_2 & 0 \end{matrix}\right].

The cross product x×y is then given by Tx(y).

[edit] Relation to the octonions

Just as the 3-dimensional cross product can be expressed in terms of the quaternions, the 7-dimensional cross product can be expressed in terms of the octonions. After identifying R7 with the imaginary octonions (the orthogonal complement of the identity in O), the cross product is given in terms of octonion multiplication by

\mathbf x \times \mathbf y = \mathrm{Im}(\mathbf{xy}) = \frac{1}{2}(\mathbf{xy}-\mathbf{yx}).

Conversely, suppose V is a 7-dimensional Euclidean space with a given cross product. Then one can define a bilinear multiplication on R⊕V as follows:

(a,\mathbf{x})(b,\mathbf{y}) = (ab - \mathbf{x}\cdot\mathbf{y}, a\mathbf y + b\mathbf x + \mathbf{x}\times\mathbf{y}).

The space R⊕V with this multiplication is then isomorphic to the octonions.

The cross product only exists in dimensions 3 and 7 since one can always define a multiplication on a space of one higher dimension as above, and this space can be shown to be a normed division algebra. Such algebras only exist in dimensions 1, 2, 4, and 8. (There is, of course, a trivial cross product is dimension 1 coming from the complex numbers; trivial since the complex numbers are commutative).

The failure of the 7-dimension cross product to satisfy the Jacobi identity is due to the nonassociativity of the octonions. In fact,

\mathbf{x}\times(\mathbf{y}\times\mathbf{z}) + \mathbf{y}\times(\mathbf{z}\times\mathbf{x}) + \mathbf{z}\times(\mathbf{x}\times\mathbf{y}) = -\frac{3}{2}[\mathbf x, \mathbf y, \mathbf z].

where [x, y, z] is the associator.

[edit] References

  • Brown, Robert B.; Gray, Alfred (1967). "Vector cross products". Commentarii Mathematici Helvetici 42 (1): 222–236. doi:10.1007/BF02564418. 
  • Lounesto, Pertti (2001). Clifford algebras and spinors. Cambridge, UK: Cambridge University Press. ISBN 0-521-00551-5. 
  • Silagadze, Z.K. (2002). "Multi-dimensional vector product". . arXiv:math.RA/0204357