Seven-dimensional cross product

This table, due to Cayley,[3][4] gives the product of orthonormal basis vectors ei and ej for each i, j from 1 to 7.

There are 480 such tables for any given set of orthogonal basis vectors, one for each of the products satisfying the definition such that each entry in the table can be expressed in terms of a single element of the basis.

is the Levi-Civita symbol, a completely antisymmetric tensor with a positive value +1 when ijk = 123, 145, 176, 246, 257, 347, 365.

The top left 3 × 3 corner of this table gives the cross product in three dimensions.

[10] Given the properties of bilinearity, orthogonality and magnitude, a nonzero cross product exists only in three and seven dimensions.

[2][11][10] This can be shown by postulating the properties required for the cross product, then deducing an equation which is only satisfied when the dimension is 0, 1, 3 or 7.

and any vector v of magnitude |v| = |x||y| sin θ in the five-dimensional space perpendicular to the plane spanned by x and y, it is possible to find a cross product with a multiplication table (and an associated set of basis vectors) such that x × y = v. Unlike in three dimensions, x × y = a × b does not imply that a and b lie in the same plane as x and y.

Because the Jacobi identity fails in seven dimensions, the seven-dimensional cross product does not give

Once we have established a multiplication table, it is then applied to general vectors x and y by expressing x and y in terms of the basis and expanding x × y through bilinearity.

Using e1 to e7 for the basis vectors a different multiplication table from the one in the Introduction, leading to a different cross product, is given with anticommutativity by[11] More compactly this rule can be written as with i = 1, ..., 7 modulo 7 and the indices i, i + 1 and i + 3 allowed to permute evenly.

The result is: As the cross product is bilinear, the operator x × – can be written as a matrix, which takes the form[citation needed] The cross product is then given by Two different multiplication tables have been used in this article, and there are more.

The numbers under the Fano diagrams (the set of lines in the diagram) indicate a set of indices for seven independent products in each case, interpreted as ijk → ei × ej = ek.

The multiplication table is recovered from the Fano diagram by following either the straight line connecting any three points, or the circle in the center, with a sign as given by the arrows.

For example, the first row of multiplications resulting in e1 in the above listing is obtained by following the three paths connected to e1 in the lower Fano diagram: the circular path e2 × e4, the diagonal path e3 × e7, and the edge path e6 × e1 = e5 rearranged using one of the above identities as: or also obtained directly from the diagram with the rule that any two unit vectors on a straight line are connected by multiplication to the third unit vector on that straight line with signs according to the arrows (sign of the permutation that orders the unit vectors).

[15] The product can also be calculated using geometric algebra of a seven-dimensional vector space with a positive-definite quadratic form.

In three dimensions, up to a scale factor there is only one trivector, the pseudoscalar of the space, and a product of the above bivector and one of the two unit trivectors gives the vector result, the dual of the bivector.

After identifying R7 with the imaginary octonions (the orthogonal complement of the real part of O), the cross product is given in terms of octonion multiplication by Conversely, suppose V is a 7-dimensional Euclidean space with a given cross product.

[18][19] The failure of the 7-dimension cross product to satisfy the Jacobi identity is related to the nonassociativity of the octonions.

[20][21] We require the product to be multi-linear, alternating, vector-valued, and orthogonal to each of the input vectors ai.

The magnitude of the product should equal the volume of the parallelotope with the vectors as edges, which can be calculated using the Gram determinant.

The conditions are The Gram determinant is the squared volume of the parallelotope with a1, ..., ak as edges.

As noted already, a binary product only exists in 7, 3, 1 and 0 dimensions, the last two being identically zero.

A further trivial 'product' arises in even dimensions, which takes a single vector and produces a vector of the same magnitude orthogonal to it through the left contraction with a suitable bivector.

As a further generalization, we can loosen the requirements of multilinearity and magnitude, and consider a general continuous function Vd → V (where V is Rn endowed with the Euclidean inner product and d ≥ 2), which is only required to satisfy the following two properties: Under these requirements, the cross product only exists (I) for n = 3, d = 2, (II) for n = 7, d = 2, (III) for n = 8, d = 3, and (IV) for any d = n − 1.

In fact this result has been generalized still further, e.g. by working over any commutative ring in which 2 is cancellable, meaning that 2x = 2y implies x = y.