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In
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 seven-dimensional cross product is a bilinear operation on
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
s in seven-dimensional Euclidean space. It assigns to any two vectors a, b in a vector also in . Like the
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 ...
in three dimensions, the seven-dimensional product is
anticommutative In mathematics, anticommutativity is a specific property of some non- commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswappe ...
and is orthogonal both to a and to b. Unlike in three dimensions, it does not satisfy the
Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
, 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
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s as the three-dimensional product does to the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s. The seven-dimensional cross product is one way of generalizing the cross product to 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. In other dimensions there are vector-valued products of three or more vectors that satisfy these conditions, and binary products with
bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector ca ...
results.


Multiplication table

The product can be given by a multiplication table, such as the one here. This table, due to Cayley, gives the product of orthonormal basis vectors e''i'' and e''j'' for each ''i'', ''j'' from 1 to 7. For example, from the table :\mathbf_1 \times \mathbf_2 = \mathbf_3 =-\mathbf_2 \times \mathbf_1 The table can be used to calculate the product of any two vectors. For example, to calculate the e1 component of x × y the basis vectors that multiply to produce e1 can be picked out to give :\left( \mathbf\right)_1 = x_2y_3 - x_3y_2 +x_4y_5-x_5y_4 + x_7y_6-x_6y_7. This can be repeated for the other six components. There are 480 such tables, one for each of the products satisfying the definition. This table can be summarized by the relation :\mathbf_i \mathbf \mathbf_j = \varepsilon _ \mathbf_k, where \varepsilon _ is 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.


Definition

The cross product on a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
''V'' is a
bilinear map In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, W ...
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 Mappings are restricted to be bilinear by and . *
orthogonality In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
: ::\mathbf \cdot (\mathbf \times \mathbf) = (\mathbf \times \mathbf) \cdot \mathbf=0, * magnitude: ::, \mathbf \times \mathbf, ^2 = , \mathbf, ^2 , \mathbf, ^2 - (\mathbf \cdot \mathbf)^2 where (x·y) is the Euclidean
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 alg ...
and , x, is the
Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
. 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 In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
''θ'' between the vectors is :, \mathbf \times \mathbf, = , \mathbf, , \mathbf, \sin \theta, which is the area of the
parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) 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 eq ...
in the plane of x and y with the two vectors as sides. A third statement of the magnitude condition is : , \mathbf \times \mathbf, = , \mathbf, , \mathbf, ~\mbox \ \left( \mathbf \cdot \mathbf \right)= 0, 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. Lounesto, pp. 96–97 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. In zero dimensions there is only the zero vector, while in one dimension all vectors are parallel, so in both these cases the product must be identically zero. The restriction to 0, 1, 3 and 7 dimensions is related to Hurwitz's theorem, that normed division algebras are only possible in 1, 2, 4 and 8 dimensions. The cross product is formed from the product of the normed division algebra by restricting it to the 0, 1, 3, or 7 imaginary dimensions of the algebra, giving nonzero products in only 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 \isin \mathbb^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. Further properties follow from the definition, including the following identities: # Anticommutativity: #: \mathbf \times \mathbf = -\mathbf \times \mathbf # Scalar triple product: #: \mathbf \cdot (\mathbf \times \mathbf) = \mathbf \cdot (\mathbf \times \mathbf) = \mathbf \cdot (\mathbf \times \mathbf) # Malcev identity: #: (\mathbf \times \mathbf) \times (\mathbf \times \mathbf) = ((\mathbf \times \mathbf) \times \mathbf) \times \mathbf + ((\mathbf \times \mathbf) \times \mathbf) \times \mathbf + ((\mathbf \times \mathbf) \times \mathbf) \times \mathbf #: \mathbf \times (\mathbf \times \mathbf) = -, \mathbf, ^2 \mathbf + (\mathbf \cdot \mathbf) \mathbf. Other properties follow only in the three-dimensional case, and are not satisfied by the seven-dimensional cross product, notably, # Vector triple product: #: \mathbf \times (\mathbf \times \mathbf) = (\mathbf \cdot \mathbf) \mathbf - (\mathbf \cdot \mathbf) \mathbf #
Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
: #: \mathbf \times (\mathbf \times \mathbf) + \mathbf \times (\mathbf \times \mathbf) + \mathbf \times (\mathbf \times \mathbf) \ne 0 Because the Jacobi Identity is not satisfied, the seven-dimensional cross product does not give R7 the structure of a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
.


Coordinate expressions

To define a particular cross product, an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
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. 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. 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 :\mathbf_1 \times \mathbf_2 = \mathbf_4, \quad \mathbf_2 \times \mathbf_4 = \mathbf_1, \quad \mathbf_4 \times \mathbf_1 = \mathbf_2, :\mathbf_2 \times \mathbf_3 = \mathbf_5, \quad \mathbf_3 \times \mathbf_5 = \mathbf_2, \quad \mathbf_5 \times \mathbf_2 = \mathbf_3, :\mathbf_3 \times \mathbf_4 = \mathbf_6, \quad \mathbf_4 \times \mathbf_6 = \mathbf_3, \quad \mathbf_6 \times \mathbf_3 = \mathbf_4, :\mathbf_4 \times \mathbf_5 = \mathbf_7, \quad \mathbf_5 \times \mathbf_7 = \mathbf_4, \quad \mathbf_7 \times \mathbf_4 = \mathbf_5, :\mathbf_5 \times \mathbf_6 = \mathbf_1, \quad \mathbf_6 \times \mathbf_1 = \mathbf_5, \quad \mathbf_1 \times \mathbf_5 = \mathbf_6, :\mathbf_6 \times \mathbf_7 = \mathbf_2, \quad \mathbf_7 \times \mathbf_2 = \mathbf_6, \quad \mathbf_2 \times \mathbf_6 = \mathbf_7, :\mathbf_7 \times \mathbf_1 = \mathbf_3, \quad \mathbf_1 \times \mathbf_3 = \mathbf_7, \quad \mathbf_3 \times \mathbf_7 = \mathbf_1. More compactly this rule can be written as : \mathbf_i \times \mathbf_ = \mathbf_ with ''i'' = 1...7
modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is ...
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 Consequence may refer to: * Logical consequence, also known as a ''consequence relation'', or ''entailment'' * In operant conditioning, a result of some behavior * Consequentialism, a theory in philosophy in which the morality of an act is determi ...
, : \mathbf_i \times \left( \mathbf_i \times \mathbf_\right) =-\mathbf_ = \mathbf_i \times \mathbf_ \ , which produces diagonals further out, and so on. The ej component of cross product x × y is given by selecting all occurrences of ej in the table and collecting the corresponding components of x from the left column and of y from the top row. The result is: :\begin\mathbf \times \mathbf = (x_2y_4 - x_4y_2 + x_3y_7 - x_7y_3 + x_5y_6 - x_6y_5)\,&\mathbf_1 \\ + (x_3y_5 - x_5y_3 + x_4y_1 - x_1y_4 + x_6y_7 - x_7y_6)\,&\mathbf_2 \\ + (x_4y_6 - x_6y_4 + x_5y_2 - x_2y_5 + x_7y_1 - x_1y_7)\,&\mathbf_3 \\ + (x_5y_7 - x_7y_5 + x_6y_3 - x_3y_6 + x_1y_2 - x_2y_1)\,&\mathbf_4 \\ + (x_6y_1 - x_1y_6 + x_7y_4 - x_4y_7 + x_2y_3 - x_3y_2)\,&\mathbf_5 \\ + (x_7y_2 - x_2y_7 + x_1y_5 - x_5y_1 + x_3y_4 - x_4y_3)\,&\mathbf_6 \\ + (x_1y_3 - x_3y_1 + x_2y_6 - x_6y_2 + x_4y_5 - x_5y_4)\,&\mathbf_7. \end As the cross product is bilinear the operator x×– can be written as a matrix, which takes the form :T_ = \begin 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. The cross product is then given by :\mathbf \times \mathbf = T_ \mathbf.


Different multiplication tables

Two different multiplication tables have been used in this article, and there are more. Further discussion of the tables and the connection of the Fano plane to these tables is found here: These multiplication tables are characterized by the Fano plane, 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: :\mathbf_6 \times \left( \mathbf_6 \times \mathbf_1 \right) = -\mathbf_1 = \mathbf_6 \times \mathbf_5 , or : \mathbf_5 \times \mathbf_6 =\mathbf_1 , 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. Considering all possible permutations of the basis there are 480 multiplication tables and so 480 cross products like this. Available a
ArXive preprint
Figure 1 is locate
here


Using geometric algebra

The product can also be calculated using
geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ...
. The product starts with the
exterior product In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of th ...
, a
bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector ca ...
valued product of two vectors: :\mathbf = \mathbf \wedge \mathbf = \frac(\mathbf - \mathbf). 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 multilinear algebra, a multivector, sometimes called Clifford number, is an element of the exterior algebra of a vector space . This algebra is graded, associative and alternating, and consists of linear combinations of simple -vectors ( ...
. 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 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 :\mathbf = \mathbf_ + \mathbf_ + \mathbf_ + \mathbf_ + \mathbf_ + \mathbf_ + \mathbf_. This is combined with the exterior product to give the cross product : \mathbf \times \mathbf = -(\mathbf \wedge \mathbf) ~\lrcorner~ \mathbf where \lrcorner 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 In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s, the 7-dimensional cross product can be expressed in terms of the
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s. After identifying \mathbb^7 with the imaginary octonions (the
orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...
of the real line in \mathbb), the cross product is given in terms of octonion multiplication by :\mathbf x \times \mathbf y = \mathrm(\mathbf) = \frac(\mathbf-\mathbf). Conversely, suppose ''V'' is a 7-dimensional Euclidean space with a given cross product. Then one can define a bilinear multiplication on \mathbb \oplus V as follows: :(a,\mathbf)(b,\mathbf) = (ab - \mathbf\cdot\mathbf, a\mathbf y + b\mathbf x + \mathbf\times\mathbf). The space \mathbb \oplus 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. 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. See also: The failure of the 7-dimension cross product to satisfy the Jacobi identity is due to the nonassociativity of the octonions. In fact, :\mathbf\times(\mathbf\times\mathbf) + \mathbf\times(\mathbf\times\mathbf) + \mathbf\times(\mathbf\times\mathbf) = -\frac mathbf x, \mathbf y, \mathbf z/math> where ''x, y, zis the
associator In abstract algebra, the term associator is used in different ways as a measure of the associativity, non-associativity of an algebraic structure. Associators are commonly studied as triple systems. Ring theory For a non-associative ring or non-as ...
.


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 G2, a subgroup of SO(7).


Generalizations

Nonzero binary cross products exist only in three and seven dimensions. Further products are possible when lifting the restriction that it must be a binary product.Lounesto, §7.5: ''Cross products of k vectors in \mathbb^n'', p. 98 We require the product to be multi-linear, alternating, vector-valued, and orthogonal to each of the input vectors a''i''. 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 In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product G_ = \left\langle v_i, v_j \right\ ...
. The conditions are *orthogonality: \left( \mathbf_1 \times \ \cdots \ \times \mathbf_k\right) \cdot \mathbf_i = 0 for i = 1,\ \dots\ , k. *the Gram determinant: , \mathbf_1 \times \cdots \times \mathbf_k , ^2 = \det (\mathbf_i \cdot \mathbf_j) = \begin \mathbf_1 \cdot \mathbf_1 & \mathbf_1 \cdot \mathbf_2 & \cdots & \mathbf_1 \cdot \mathbf_k\\ \mathbf_2 \cdot \mathbf_1 & \mathbf_2 \cdot \mathbf_2 & \cdots & \mathbf_2 \cdot \mathbf_k\\ \vdots & \vdots & \ddots & \vdots \\ \mathbf_k \cdot \mathbf_1 & \mathbf_k \cdot \mathbf_2 & \cdots & \mathbf_k \cdot \mathbf_k\\ \end The Gram determinant is the squared volume of the parallelotope with a1, ..., a''k'' 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 ''n'' − 1 vectors in ''n'' ≥ 3 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 \mathbf \times \mathbf \times \mathbf = (\mathbf \wedge \mathbf \wedge \mathbf) ~\lrcorner~ (\mathbf - \mathbf_8) where v is the same trivector as used in seven dimensions, \lrcorner 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 V^d \to V (where V is \mathbb^n endowed with the Euclidean inner product and d \geq 2 ) which is only required to satisfy the following two properties: # The cross product is always orthogonal to all the input vectors. # If the input vectors are linearly independent, then the cross product is nonzero. Under these requirements, the cross product only exists (I) for n = 3, d = 2, (II) for n = 7, d = 3, (III) for n = 8, d = 3, and (IV) for any d = n - 1 .


See also

* Composition algebra


Notes


References

* * * Also available as ArXiv reprint . * {{DEFAULTSORT:Seven-Dimensional Cross Product Bilinear maps Octonions Operations on vectors