In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the seven-dimensional cross product is a bilinear operation on vectors in a seven-dimensional space

Seven-dimensional space

In physics and mathematics, a sequence of n numbers can also be understood as a location in n-dimensional space. When n = 7, the set of all such locations is called 7-dimensional Euclidean space...

. It assigns to any two vectors a, b in ℝ⁷ a vector a × b ∈ ℝ⁷. In seven dimensions there also exists a cross product involving six vectors (which is linear but not binary), discussed briefly in the section on generalizations.

As in the more familiar three-dimensional cross product

Cross product

In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...

, the binary cross product in seven dimensions is alternating and orthogonal to the original vectors, but unlike that case, however, it does not satisfy the Jacobi identity

Jacobi identity

In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. Unlike for associative operations, order of evaluation is significant for operations satisfying Jacobi identity...

. The seven dimensional cross product has the same relationship to octonion

Octonion

In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H...

s as the three-dimensional cross product does to quaternion

Quaternion

In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...

s, and apart from the trivial cases of zero and one dimensions, binary cross products can be shown to exist only in three and seven dimensions.

Example

The postulates underlying construction of the seven-dimensional cross product are presented in the section Definition. As context for that discussion, the historically first example of the cross product is tabulated below using e₁ to e₇ as basis vectors. This table is one of 480 independent multiplication tables fitting the pattern that each unit vector appears once in each column and once in each row. Thus, each unit vector appears as a product in the table six times, three times with a positive sign and three with a negative sign because of antisymmetry about the diagonal of zero entries. For example, e₁ = e₂ × e₃ = e₄ × e₅ = e₇ × e₆ and the negative entries are the reversed cross-products.

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align="bottom" style="caption-side: bottom"| Cayley's sample multiplication table

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Entries in the interior give the product of the corresponding vectors on the left and the top in that order (the product is anti-commutative). Some entries are highlighted to emphasize the symmetry.

The table can be summarized by the relation


where is a completely antisymmetric tensor with a positive value +1 when ijk = 123, 145, 176, 246, 257, 347, 365. By picking out the factors leading to the unit vector e₁, for example, one finds the formula for the e₁ component of x × y. Namely

The top left 3 × 3 corner of the table is the same as the cross product in three dimensions. It also may be noticed that orthogonality of the cross product to its constituents x and y is a requirement upon the entries in this table. However, because of the many possible multiplication tables, general results for the cross product are best developed using a basis-independent formulation, as introduced next.

Definition

We can define a cross product on a Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 V as a bilinear map  from V × V to V mapping vectors x and y in V to another vector x × y also in V, where x × y has the properties
   (orthogonality),

and:
   (magnitude),

where (x·y) is the Euclidean dot product

Dot product

In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

 and |x| is the vector norm

Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

. The first property states that the cross product is perpendicular to its arguments, while the second property gives the magnitude of the cross product. An equivalent expression in terms of the angle

Angle

In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

 θ between the vectors is


or the area of the parallelogram

Parallelogram

In Euclidean geometry, a parallelogram is a convex quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure...

 in the plane of x and y with the two vectors as sides. As a third alternative the following can be shown to be equivalent to either expression for the magnitude:

Consequences of the defining properties

Given the three basic properties of (i) bilinearity, (ii) orthogonality and (iii) magnitude discussed in the section on definition, a nontrivial cross product exists only in three and seven dimensions. This restriction upon dimensionality can be shown by postulating the properties required for the cross product, then deducing a equation which is only satisfied when the dimension is 0, 1, 3 or 7. In zero dimensions there is only the zero vector, while in one dimension all vectors are parallel, so in both these cases a cross product must be identically zero.

The restriction to 0, 1, 3 and 7 dimensions is related to Hurwitz's theorem, that normed division algebra

Normed division algebra

In mathematics, a normed division algebra A is a division algebra over the real or complex numbers which is also a normed vector space, with norm || · || satisfying the following property:\|xy\| = \|x\| \|y\| for all x and y in A....

s are only possible in 1, 2, 4 and 8 dimensions. The cross product is derived from the product of the algebra by considering the product restricted to the 0, 1, 3, or 7 imaginary dimensions of the algebra. Again discarding trivial products the product can only be defined this way in three and seven dimensions.

In contrast with three dimensions where the cross product 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 ∈ ℝ⁷ and any vector v of magnitude |v| = |x||y| sinθ 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. That leaves open the question of just how many vector pairs like x and y can be matched to specified directions like v before the limitations of any particular table intervene.

Another difference between the three dimensional cross product and a seven dimensional cross product is:
This statement is exemplified by every multiplication table, because any specific unit vector selected as a product occurs as a mapping from three different pairs of unit vectors, once with a plus sign and once with a minus sign. Each of these different pairs, of course, corresponds to another plane being mapped into the same direction.

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

Anticommutativity

In mathematics, anticommutativity is the property of an operation that swapping the position of any two arguments negates the result. Anticommutative operations are widely used in algebra, geometry, mathematical analysis and, as a consequence, in physics: they are often called antisymmetric...

),
  (scalar triple product),
  (Malcev identity

Malcev algebra

In mathematics, a Malcev algebra over a field is a nonassociative algebra that is antisymmetric, so thatxy = -yx\ and satisfies the Malcev identity = x + x + y.\...

),


Other properties follow only in the three dimensional case, and are not satisfied by the seven dimensional cross product, notably,
  (vector triple product),
   (Jacobi identity

Jacobi identity

In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. Unlike for associative operations, order of evaluation is significant for operations satisfying Jacobi identity...

).

Coordinate expressions

To define a particular cross product, an orthonormal basis

Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...

 {e_j} may be selected and a multiplication table provided that determines all the products {e_i× e_j}. One possible multiplication table is described in the Example section, but it is not unique. Unlike three dimensions, there are many tables because every pair of unit vectors is perpendicular to five other unit vectors, allowing many choices for each cross product.

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. !style=background:#FAEBD12 |×
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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








More compactly this rule can be written as

with i = 1...7 modulo

Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....

 7 and the indices i, i + 1 and i + 3 allowed to permute evenly. Together with anticommutativity this generates the product. This rule directly produces the two diagonals immediately adjacent to the diagonal of zeros in the table. Also, from an identity in the subsection on consequences,

which produces diagonals further out, and so on.

The e_j component of cross product x × y is given by selecting all occurrences of e_j in the table and collecting the corresponding components of x from the left column and of y from the top row. The result is:


As the cross product is bilinear the operator x×– can be written as a matrix, which takes the form


The cross product is then given by

Different multiplication tables

Two different multiplication tables have been used in this article, and there are more. These multiplication tables are characterized by the Fano plane

Fano plane

In finite geometry, the Fano plane is the finite projective plane with the smallest possible number of points and lines: 7 each.-Homogeneous coordinates:...

, and these are shown in the figure for the two tables used here: at top, the one described by Sabinin, Sbitneva, and Shestakov, and at bottom that described by Lounesto. 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 → e_i × e_j = e_k. 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).

It can be seen that both multiplication rules follow from the same Fano diagram by simply renaming the unit vectors, and changing the sense of the center unit vector. The question arises: how many multiplication tables are there?

Using geometric algebra

The product can also be calculated using geometric algebra

Geometric algebra

Geometric algebra , together with the associated Geometric calculus, provides a comprehensive alternative approach to the algebraic representation of classical, computational and relativistic geometry. GA now finds application in all of physics, in graphics and in robotics...

. The product starts with the exterior product, a bivector

Bivector

In mathematics, a bivector or 2-vector is a quantity in geometric algebra or exterior algebra that generalises the idea of a vector. If a scalar is considered a zero dimensional quantity, and a vector is a one dimensional quantity, then a bivector can be thought of as two dimensional. Bivectors...

 valued product of two vectors:


This is bilinear, alternate, has the desired magnitude, but is not vector valued. The vector, and so the cross product, comes from the product of this bivector with a trivector. In three dimensions up to a scale factor there is only one trivector, the pseudoscalar

Pseudoscalar

In physics, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion such as improper rotations while a true scalar does not.The prototypical example of a pseudoscalar is the scalar triple product...

 of the space, and a product of the above bivector and one of the two unit trivectors gives the vector result, the dual

Hodge dual

In mathematics, the Hodge star operator or Hodge dual is a significant linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented inner product space.-Dimensions and algebra:...

 of the bivector.

A similar calculation is done is seven dimensions, except as 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


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


where 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 quaternion

Quaternion

In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...

s, the 7-dimensional cross product can be expressed in terms of the octonion

Octonion

In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H...

s. After identifying R⁷ with the imaginary octonions (the orthogonal complement of the real line in 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. Then one can define a bilinear multiplication on R⊕V as follows:
The space R⊕V 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

Normed division algebra

In mathematics, a normed division algebra A is a division algebra over the real or complex numbers which is also a normed vector space, with norm || · || satisfying the following property:\|xy\| = \|x\| \|y\| for all x and y in A....

. 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 due to the nonassociativity of the octonions. In fact,
where [x, y, z] is the associator

Associator

In abstract algebra, the term associator is used in different ways as a measure of the nonassociativity of an algebraic structure.-Ring theory:...

.

Rotations

In three dimensions the cross product is invariant under the group of the rotation group, SO(3)

Rotation group

In mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves orientation ...

, 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)

Orthogonal group

In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...

. Instead it is invariant under the exceptional Lie group G₂, a subgroup of SO(7).

Generalizations

Non-trivial binary cross products exist only in three and seven dimensions. But if the restriction that the product is binary is lifted, so products of more than two vectors are allowed, then more products are possible. As in two dimensions the product must be vector valued, linear, and anti-commutative in any two of the vectors in the product.

The product should satisfy orthogonality, so it is orthogonal to all its members. This means no more than n - 1 vectors can be used in n dimensions. The magnitude of the product should equal the volume of the parallelotope with the vectors as edges, which is can be calculated using the Gram determinant. So the conditions are
(orthogonality) (Gram determinant)

The Gram determinant is the squared volume of the parallelotope with a₁, ..., a_k as edges. If there are just two vectors x and y it simplifies to the condition for the binary cross product given above, that is
,

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

• as a binary product in three and seven dimensions
• as a product of n - 1 vectors in n > 3 dimensions
• as a product of three vectors in eight dimensions

The product of n - 1 vectors is in n dimensions is the Hodge dual

Hodge dual

In mathematics, the Hodge star operator or Hodge dual is a significant linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented inner product space.-Dimensions and algebra:...

of the exterior product of n - 1 vectors. One version of the product of three vectors in eight dimensions is given by

where v is the same trivector as used in seven dimensions, ⌋ is again the left contraction, and w = -ve_12...7 is a 4-vector.

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