In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space. It assigns to any two vectors a, b in a vector also in . Like the cross product in three dimensions, the seven-dimensional product is anticommutative and is orthogonal both to a and to b. Unlike in three dimensions, it does not satisfy the Jacobi identity, and while the three-dimensional cross product is unique up to a sign, there are many seven-dimensional cross products. The seven-dimensional cross product has the same relationship to the octonions as the three-dimensional product does to the quaternions.

The seven-dimensional cross product is one way of generalizing the cross product to dimensions other than three dimensions, and it is the only other bilinear product of two vectors that is vector-valued, orthogonal, and has the same magnitude as in the 3D case. gives the product of orthonormal basis vectors e<sub>i</sub> and e<sub>j</sub> for each i, j from 1 to 7. For example, from the table

: <math>\mathbf{e}_1 \times \mathbf{e}_2 = \mathbf{e}_3 =-\mathbf{e}_2 \times \mathbf{e}_1</math>

The table can be used to calculate the product of any two vectors. For example, to calculate the e<sub>1</sub> component of x × y the basis vectors that multiply to produce e<sub>1</sub> can be picked out to give

: <math>\left( \mathbf{ x \times y}\right)_1 = x_2y_3 - x_3y_2 +x_4y_5-x_5y_4 + x_7y_6-x_6y_7.</math>

This can be repeated for the other six components.

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. This table can be summarized by the relation

  • orthogonality:
  • : <math>\mathbf{x} \cdot (\mathbf{x} \times \mathbf{y}) = (\mathbf{x} \times \mathbf{y}) \cdot \mathbf{y}=0,</math>
  • magnitude:
  • : <math>|\mathbf{x} \times \mathbf{y}|^2 = |\mathbf{x}|^2 |\mathbf{y}|^2 - (\mathbf{x} \cdot \mathbf{y})^2 </math>

where (x·y) is the Euclidean dot product and |x| is the Euclidean norm. The first property states that the product is perpendicular to its arguments, while the second property gives the magnitude of the product. An equivalent expression in terms of the angle θ between the vectors is

: <math>|\mathbf{x} \times \mathbf{y}| = |\mathbf{x}| |\mathbf{y}| \sin \theta, </math>

which is the area of the parallelogram in the plane of x and y with the two vectors as sides. A third statement of the magnitude condition is

: <math>|\mathbf{x} \times \mathbf{y}| = |\mathbf{x}| |\mathbf{y}|~\mbox{if} \ \left( \mathbf{x} \cdot \mathbf{y} \right)= 0,</math>

if x × x = 0 is assumed as a separate axiom.

Consequences of the defining properties

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

In contrast to the three-dimensional cross product, which is unique (apart from sign), there are many possible binary cross products in seven dimensions. One way to see this is to note that given any pair of vectors x and y <math>\isin \mathbb{R}^7</math> 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.

Further properties follow from the definition, including the following identities:

  1. Anticommutativity:
  2. : <math> \mathbf{x} \times \mathbf{y} = -\mathbf{y} \times \mathbf{x} </math>
  3. Scalar triple product:
  4. : <math> \mathbf{x} \cdot (\mathbf{y} \times \mathbf{z}) = \mathbf{y} \cdot (\mathbf{z} \times \mathbf{x}) = \mathbf{z} \cdot (\mathbf{x} \times \mathbf{y})</math>
  5. Malcev identity:
  6. : <math> (\mathbf{x} \times \mathbf{y}) \times (\mathbf{x} \times \mathbf{z}) = ((\mathbf{x} \times \mathbf{y}) \times \mathbf{z}) \times \mathbf{x} + ((\mathbf{y} \times \mathbf{z}) \times \mathbf{x}) \times \mathbf{x} + ((\mathbf{z} \times \mathbf{x}) \times \mathbf{x}) \times \mathbf{y}</math>
  7. : <math> \mathbf{x} \times (\mathbf{x} \times \mathbf{y}) = -|\mathbf{x}|^2 \mathbf{y} + (\mathbf{x} \cdot \mathbf{y}) \mathbf{x}.</math>

Other properties follow only in the three-dimensional case, and are not satisfied by the seven-dimensional cross product, notably,

  1. Vector triple product:
  2. : <math> \mathbf{x} \times (\mathbf{y} \times \mathbf{z}) = (\mathbf{x} \cdot \mathbf{z}) \mathbf{y} - (\mathbf{x} \cdot \mathbf{y}) \mathbf{z} </math>
  3. Jacobi identity:
  4. : <math> \mathbf{x} \times (\mathbf{y} \times \mathbf{z}) + \mathbf{y} \times (\mathbf{z} \times \mathbf{x}) + \mathbf{z} \times (\mathbf{x} \times \mathbf{y}) = 0</math>

Thanks to the Jacobi Identity, the three-dimensional cross product gives <math>\mathbb{R}^3</math> the structure of a Lie algebra, which is isomorphic to <math>\mathfrak{so}(3)</math>, the Lie algebra of the 3d rotation group. Because the Jacobi identity fails in seven dimensions, the seven-dimensional cross product does not give <math>\mathbb{R}^7</math> the structure of a Lie algebra.

Coordinate expressions

To define a particular cross product, an orthonormal basis may be selected and a multiplication table provided that determines all the products . One possible multiplication table is described in the Multiplication table section, but it is not unique. These multiplication tables are characterized by the Fano plane,

Using geometric algebra

The product can also be calculated using geometric algebra of a seven-dimensional vector space with a positive-definite quadratic form. The product starts with the exterior product, a bivector-valued product of two vectors:

: <math>\mathbf{B} = \mathbf{x} \wedge \mathbf{y} = \frac{1}{2}(\mathbf{xy} - \mathbf{yx}).</math>

This is bilinear, alternating, has the desired magnitude, but is not vector-valued. The vector, and so the cross product, comes from the contraction of this bivector with a trivector. 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.

A similar calculation is done in seven dimensions, except that since trivectors form a 35-dimensional space there are many trivectors that could be used, though not just any trivector will do. The trivector that gives the same product as the above coordinate transform is

: <math>\mathbf{v} = \mathbf{e}_{124} + \mathbf{e}_{235} + \mathbf{e}_{346} + \mathbf{e}_{457} + \mathbf{e}_{561} + \mathbf{e}_{672} + \mathbf{e}_{713}.</math>

This is combined with the exterior product to give the cross product

: <math> \mathbf{x} \times \mathbf{y} = -(\mathbf{x} \wedge \mathbf{y}) ~\lrcorner~ \mathbf{v} </math>

where <math> \lrcorner </math> is the left contraction operator from geometric algebra.

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 <math>\mathbb{R}</math><sup>7</sup> with the imaginary octonions (the orthogonal complement of the real part of <math>\mathbb{O}</math>), the cross product is given in terms of octonion multiplication by

: <math>\mathbf x \times \mathbf y = \mathrm{Im}(\mathbf{xy}) = \frac{1}{2}(\mathbf{xy}-\mathbf{yx}).</math>

Conversely, suppose <math>V </math> is a 7-dimensional Euclidean space with a given cross product. Then one can define a bilinear multiplication on <math>\mathbb{R}\oplus V</math> as follows:

: <math>(a,\mathbf{x})(b,\mathbf{y}) = (ab - \mathbf{x}\ast\mathbf{y}, a\mathbf y + b\mathbf x + \mathbf{x}\times\mathbf{y}).</math>

The space <math>\mathbb{R}\oplus V</math> with this multiplication is then isomorphic to the octonions.

The cross product only exists in three and seven dimensions as 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. By Hurwitz's theorem such algebras only exist in one, two, four, and eight dimensions, so the cross product must be in zero, one, three or seven dimensions. The products in zero and one dimensions are trivial, so non-trivial cross products only exist in three and seven dimensions.

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

: <math>\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]</math>

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

Rotations

In three dimensions the cross product is invariant under the action of the rotation group, SO(3), so the cross product of x and y after they are rotated is the image of under the rotation. But this invariance is not true in seven dimensions; that is, the cross product is not invariant under the group of rotations in seven dimensions, SO(7). Instead it is invariant under the exceptional Lie group G<sub>2</sub>, a subgroup of SO(7). We require the product to be multi-linear, alternating, vector-valued, and orthogonal to each of the input vectors a<sub>i</sub>. The orthogonality requirement implies that in n dimensions, no more than vectors can be used. 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

  • orthogonality: <math display="block">\left( \mathbf{a}_1 \times \ \cdots \ \times \mathbf{a}_k\right) \cdot \mathbf{a}_i = 0</math> for .
  • the Gram determinant: <math display="block">|\mathbf{a}_1 \times \cdots \times \mathbf{a}_k |^2

= \det (\mathbf{a}_i \cdot \mathbf{a}_j) =

\begin{vmatrix}

\mathbf{a}_1 \cdot \mathbf{a}_1 & \mathbf{a}_1 \cdot \mathbf{a}_2 & \cdots & \mathbf{a}_1 \cdot \mathbf{a}_k\\

\mathbf{a}_2 \cdot \mathbf{a}_1 & \mathbf{a}_2 \cdot \mathbf{a}_2 & \cdots & \mathbf{a}_2 \cdot \mathbf{a}_k\\

\vdots & \vdots & \ddots & \vdots \\

\mathbf{a}_k \cdot \mathbf{a}_1 & \mathbf{a}_k \cdot \mathbf{a}_2 & \cdots & \mathbf{a}_k \cdot \mathbf{a}_k\\

\end{vmatrix}

</math>

The Gram determinant is the squared volume of the parallelotope with a<sub>1</sub>, ..., a<sub>k</sub> as edges.

With these conditions a non-trivial cross product only exists:

  • as a binary product in three and seven dimensions
  • as a product of vectors in dimensions, being the Hodge dual of the exterior product of the vectors
  • as a product of three vectors in eight dimensions

One version of the product of three vectors in eight dimensions is given by

<math display="block">\mathbf{a} \times \mathbf{b} \times \mathbf{c} = (\mathbf{a} \wedge \mathbf{b} \wedge \mathbf{c}) ~\lrcorner~ (\mathbf{w} - \mathbf{ve}_8)</math>

where v is the same trivector as used in seven dimensions, <math>\lrcorner</math> is again the left contraction, and is a 4-vector.

There are also trivial products. 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. In two dimensions this is a rotation through a right angle.

As a further generalization, we can loosen the requirements of multilinearity and magnitude, and consider a general continuous function (where V is R<sup>n</sup> endowed with the Euclidean inner product and ), which is only required to satisfy the following two properties:

  1. The cross product is always orthogonal to all the input vectors.
  2. If the input vectors are linearly independent, then the cross product is nonzero.

Under these requirements, the cross product only exists (I) for , (II) for , (III) for , and (IV) for any .

See also

  • Composition algebra

Notes