View source for Seven-dimensional cross product

{{short description|Mathematics concept}}
In [[Mathematics|mathematics]], the '''seven-dimensional cross product''' is a bilinear operation on vectors in [[Seven-dimensional space|seven-dimensional Euclidean space]]. It assigns to any two vectors '''a''', '''b''' in {{tmath|\mathbb{R}^7}} a vector {{nowrap|'''a''' × '''b'''}} also in {{tmath|\mathbb{R}^7}}.<ref name=Massey0>

{{cite journal |title=Cross products of vectors in higher dimensional Euclidean spaces |author=WS Massey |journal=The American Mathematical Monthly |volume=90 |year=1983 |pages=697–701 |publisher=Mathematical Association of America |jstor=2323537 |issue=10 |doi=10.2307/2323537 }}

</ref> Like the [[Cross product|cross product]] in three dimensions, the seven-dimensional product is [[Anticommutativity|anticommutative]] and {{nowrap|'''a''' × '''b'''}} 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 [[Octonion|octonion]]s as the three-dimensional product does to the [[Quaternion|quaternion]]s.

The seven-dimensional cross product is one way of generalising 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.<ref name=Massey2/> In other dimensions there are vector-valued products of three or more vectors that satisfy these conditions, and binary products with [[Bivector|bivector]] results.

==Multiplication table==

{| class="wikitable" style="float: right; text-align: center; margin:0.2em auto"
|-
 !|'''×'''
  !{{math|'''e'''<sub>1</sub>}}
  !{{math|'''e'''<sub>2</sub>}}
  !{{math|'''e'''<sub>3</sub>}}
  !{{math|'''e'''<sub>4</sub>}}
  !{{math|'''e'''<sub>5</sub>}}
  !{{math|'''e'''<sub>6</sub>}}
  !{{math|'''e'''<sub>7</sub>}}
 |-
!{{math|e<sub>1</sub>}}
  |{{math|0}}
  |{{math|'''e'''<sub>3</sub>}}
  |{{math|−'''e'''<sub>2</sub>}}
  |{{math|'''e'''<sub>5</sub>}}
  |{{math|−'''e'''<sub>4</sub>}}
  |{{math|−'''e'''<sub>7</sub>}}
  |{{math|'''e'''<sub>6</sub>}}
 |-
!{{math|e<sub>2</sub>}}
 |{{math|−'''e'''<sub>3</sub>}}
  |{{math|0}}
  |{{math|'''e'''<sub>1</sub>}}
  ||{{math|'''e'''<sub>6</sub>}}
  |{{math|'''e'''<sub>7</sub>}}
  ||{{math|−'''e'''<sub>4</sub>}}
  ||{{math|−'''e'''<sub>5</sub>}}
|-
!{{math|e<sub>3</sub>}}
  |{{math|'''e'''<sub>2</sub>}}
  |{{math|−'''e'''<sub>1</sub>}}
  |{{math|0}}
  |{{math|'''e'''<sub>7</sub>}}
  |{{math|−'''e'''<sub>6</sub>}}
  |{{math|'''e'''<sub>5</sub>}}
  |{{math|−'''e'''<sub>4</sub>}}
|-
!{{math|e<sub>4</sub>}}
  ||{{math|−'''e'''<sub>5</sub>}}
  ||{{math|−'''e'''<sub>6</sub>}}
  |{{math|−'''e'''<sub>7</sub>}}
  |{{math|0}}
  |{{math|'''e'''<sub>1</sub>}}
  |{{math|'''e'''<sub>2</sub>}}
  |{{math|'''e'''<sub>3</sub>}}
 |-
!{{math|e<sub>5</sub>}}
  ||{{math|'''e'''<sub>4</sub>}}
  |{{math|−'''e'''<sub>7</sub>}}
  |{{math|'''e'''<sub>6</sub>}}
  |{{math|−'''e'''<sub>1</sub>}}
  |{{math|0}}
  |{{math|−'''e'''<sub>3</sub>}}
  |{{math|'''e'''<sub>2</sub>}}
|-
!{{math|e<sub>6</sub>}}
  |{{math|'''e'''<sub>7</sub>}}
  |{{math|'''e'''<sub>4</sub>}}
  |{{math|−'''e'''<sub>5</sub>}}
  |{{math|−'''e'''<sub>2</sub>}}
  |{{math|'''e'''<sub>3</sub>}}
  |{{math|0}}
  |{{math|−'''e'''<sub>1</sub>}}
|-
!{{math|e<sub>7</sub>}}
  |{{math|−'''e'''<sub>6</sub>}}
  |{{math|'''e'''<sub>5</sub>}}
  |{{math|'''e'''<sub>4</sub>}}
  |{{math|−'''e'''<sub>3</sub>}}
  |{{math|−'''e'''<sub>2</sub>}}
  |{{math|'''e'''<sub>1</sub>}}
  |{{math|0}}
|-
|}

The product can be given by a multiplication table, such as the one here. This table, due to Cayley,<ref name=Cayley>{{cite book |title=Hypercomplex analysis |edition=Conference on quaternionic and Clifford analysis; proceedings |chapter-url=https://books.google.com/books?id=H-5v6pPpyb4C&pg=PA168 |page=168 |author=G Gentili, C Stoppato, DC Struppa and F Vlacci |chapter=Recent developments for regular functions of a hypercomplex variable|editor1=Irene Sabadini |editor1-link=Irene Sabadini|editor2=M Shapiro |editor3=F Sommen |isbn=978-3-7643-9892-7 |year=2009 |publisher=Birkhäuser}}
</ref><ref name= Shestakov>
{{Cite book |title=Non-associative algebra and its applications |author1=Lev Vasilʹevitch Sabinin |author2=Larissa Sbitneva |author3=I. P. Shestakov |page=235 |chapter=§17.2 Octonion algebra and its regular bimodule representation |chapter-url=https://books.google.com/books?id=_PEWt18egGgC&pg=PA235 |isbn=0-8247-2669-3 |year=2006|publisher=CRC Press }} 
</ref> 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, one for each of the products satisfying the definition.<ref name=Parra>
{{cite book |title=Clifford algebras with numeric and symbolic computations |author1=Rafał Abłamowicz |author2=Pertti Lounesto |author3=Josep M. Parra |chapter-url=https://books.google.com/books?id=OpbY_abijtwC&pg=PA202 |page=202 |chapter=§ Four octonionic basis numberings |publisher=Birkhäuser |year=1996 |isbn=0-8176-3907-1}}
</ref> This table can be summarized by the relation<ref name= Shestakov/>
:<math>\mathbf{e}_i \mathbf{\times} \mathbf{e}_j =  \varepsilon _{ijk} \mathbf{e}_k, </math>
where <math>\varepsilon _{ijk}</math> 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]] ''V'' is a [[Bilinear map|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<ref name=Massey0/><ref name=Brown>
Mappings are restricted to be bilinear by {{Harv|Massey|1993}} and {{cite journal |title=Vector cross products |author1=Robert B Brown  |author2=Alfred Gray  |name-list-style=amp |pages=222–236 |journal=Commentarii Mathematici Helvetici |volume=42  |year=1967 |issue= 1/December |doi=10.1007/BF02564418 |publisher=Birkhäuser Basel|s2cid=121135913 }}.</ref>

*'''[[Orthogonality|orthogonality]]:'''
::<math>\mathbf{x} \cdot (\mathbf{x} \times \mathbf{y}) = (\mathbf{x} \times \mathbf{y}) \cdot \mathbf{y}=0,</math>
*'''[[Norm (mathematics)|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|dot product]] and |'''x'''| is the [[Norm (mathematics)|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#Dot product and generalisation|angle]] ''θ'' between the vectors<ref name=Hildebrand>{{cite book |title=Methods of applied mathematics |author=Francis Begnaud Hildebrand |page=24 |url=https://books.google.com/books?id=17EZkWPz_eQC&pg=PA24|isbn=0-486-67002-3 |edition=Reprint of Prentice-Hall 1965 2nd|publisher=Courier Dover Publications |year=1992}}
</ref> is<ref name = Lounesto/>

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

which is the area of the [[Parallelogram|parallelogram]] in the plane of '''x''' and '''y''' with the two vectors as sides.<ref>{{cite book
|title=A Course in the Geometry of N Dimensions
|first1=M. G.
|last1=Kendall
|publisher=Courier Dover Publications
|year=2004
|isbn=0-486-43927-5
|page=19
|url=https://books.google.com/books?id=_dFJ6pSzRLkC&pg=PA19}}
</ref> 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.<ref name=Silagadze1>
{{cite journal | author = Z.K. Silagadze | title = Multi-dimensional vector product | year = 2002 | doi = 10.1088/0305-4470/35/23/310 | journal = Journal of Physics A: Mathematical and General | volume = 35 | issue = 23 | pages = 4949–4953 |arxiv=math.RA/0204357  | bibcode = 2002JPhA...35.4949S | s2cid = 119165783 }}
</ref>

==Consequences of the defining properties==
Given the properties of bilinearity, orthogonality and magnitude, a nonzero cross product exists only in three and seven dimensions.<ref name=Massey2>
{{cite journal |title=Cross products of vectors in higher dimensional Euclidean spaces |author=WS Massey |year=1983 |jstor=2323537|quote=If one requires only three basic properties of the cross product ... it turns out that a cross product of vectors exists only in 3-dimensional and 7-dimensional Euclidean space. |pages=697–701 |journal=The American Mathematical Monthly |volume=90 |issue=10 |doi=10.2307/2323537}}</ref><ref name = Lounesto>Lounesto, pp. 96–97
</ref><ref name=Silagadze1/>  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 (normed division algebras)|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.<ref name=Jacobson>
{{cite book |title= Basic algebra I |author=Nathan Jacobson |publisher=Dover Publications |year=2009 |pages=417–427 |isbn=978-0-486-47189-1 |edition=Reprint of Freeman 1974 2nd  |url=https://books.google.com/books?id=_K04QgAACAAJ }}</ref>

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'''.<ref name = Lounesto/>

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

#'''[[Anticommutativity]]:'''
#:<math> \mathbf{x} \times \mathbf{y} = -\mathbf{y} \times \mathbf{x} </math>
#'''[[Scalar triple product]]:'''
#:<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>
#'''[[Malcev algebra|Malcev identity]]:'''<ref name=Lounesto/>
#:<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>
#:<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,
#'''[[Vector triple product]]:'''
#:<math> \mathbf{x} \times (\mathbf{y} \times \mathbf{z}) \ne (\mathbf{x} \cdot \mathbf{z}) \mathbf{y} - (\mathbf{x} \cdot \mathbf{y}) \mathbf{z} </math>
#'''[[Jacobi identity]]:'''<ref name=Lounesto/>
#:<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}) \ne 0</math>

Because the Jacobi Identity is not satisfied, the seven-dimensional cross product does not give '''R'''<sup>7</sup> the structure of a [[Lie algebra]].

==Coordinate expressions==
To define a particular cross product, an [[Orthonormal basis|orthonormal basis]] {'''e'''<sub>''j''</sub>} may be selected and a multiplication table provided that determines all the products {'''e'''<sub>''i''</sub> × '''e'''<sub>''j''</sub>}. One possible multiplication table is described in the Example section, but it is not unique.<ref name=Parra/> 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. 
{| class="wikitable" style="float: right;text-align: center;" 
|-
 !style=background:#FAEBD12 |'''×'''
  !{{math|'''e'''<sub>1</sub>}}
  !{{math|'''e'''<sub>2</sub>}}
  !{{math|'''e'''<sub>3</sub>}}
  !{{math|'''e'''<sub>4</sub>}}
  !{{math|'''e'''<sub>5</sub>}}
  !{{math|'''e'''<sub>6</sub>}}
  !{{math|'''e'''<sub>7</sub>}}
 |-
 !{{math|'''e'''<sub>1</sub>}}
  ||{{math|0}}
  ||{{math|'''e'''<sub>4</sub>}}
  ||{{math|'''e'''<sub>7</sub>}}
  ||{{math|−'''e'''<sub>2</sub>}}
  ||{{math|'''e'''<sub>6</sub>}}
  ||{{math|−'''e'''<sub>5</sub>}}
  |{{math|−'''e'''<sub>3</sub>}}
 |-
 !{{math|'''e'''<sub>2</sub>}}
  ||{{math|−'''e'''<sub>4</sub>}}
  ||{{math|0}}
  ||{{math|'''e'''<sub>5</sub>}}
  ||{{math|'''e'''<sub>1</sub>}}
  ||{{math|−'''e'''<sub>3</sub>}}
  ||{{math|'''e'''<sub>7</sub>}}
  ||{{math|−'''e'''<sub>6</sub>}}
 |-
  !{{math|'''e'''<sub>3</sub>}}
  ||{{math|−'''e'''<sub>7</sub>}}
  ||{{math|−'''e'''<sub>5</sub>}}
  ||{{math|0}}
  ||{{math|'''e'''<sub>6</sub>}}
  ||{{math|'''e'''<sub>2</sub>}}
  ||{{math|−'''e'''<sub>4</sub>}}
  ||{{math|'''e'''<sub>1</sub>}}
 |-
  !{{math|'''e'''<sub>4</sub>}}
  ||{{math|'''e'''<sub>2</sub>}}
  ||{{math|−'''e'''<sub>1</sub>}}
  ||{{math|−'''e'''<sub>6</sub>}}
  ||{{math|0}}
  ||{{math|'''e'''<sub>7</sub>}}
  |{{math|'''e'''<sub>3</sub>}}
  ||{{math|−'''e'''<sub>5</sub>}}
 |-
  !{{math|'''e'''<sub>5</sub>}}
  ||{{math|−'''e'''<sub>6</sub>}}
  ||{{math|'''e'''<sub>3</sub>}}
  ||{{math|−'''e'''<sub>2</sub>}}
  ||{{math|−'''e'''<sub>7</sub>}}
  ||{{math|0}}
  ||{{math|'''e'''<sub>1</sub>}}
  ||{{math|'''e'''<sub>4</sub>}}
|-
  !{{math|'''e'''<sub>6</sub>}}
  ||{{math|'''e'''<sub>5</sub>}}
  ||{{math|−'''e'''<sub>7</sub>}}
  ||{{math|'''e'''<sub>4</sub>}}
  ||{{math|−'''e'''<sub>3</sub>}}
  ||{{math|−'''e'''<sub>1</sub>}}
  ||{{math|0}}
  ||{{math|'''e'''<sub>2</sub>}}
|-
 !{{math|'''e'''<sub>7</sub>}}
  ||{{math|'''e'''<sub>3</sub>}}
  ||{{math|'''e'''<sub>6</sub>}}
  ||{{math|−'''e'''<sub>1</sub>}}
  ||{{math|'''e'''<sub>5</sub>}}
  ||{{math|−'''e'''<sub>4</sub>}}
  ||{{math|−'''e'''<sub>2</sub>}}
  ||{{math|0}}
 |-
|}
Using '''e'''<sub>1</sub>  to '''e'''<sub>7</sub> 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<ref name="Lounesto"/>

:<math>\mathbf{e}_1 \times \mathbf{e}_2 = \mathbf{e}_4, \quad \mathbf{e}_2 \times \mathbf{e}_4 = \mathbf{e}_1, \quad \mathbf{e}_4 \times \mathbf{e}_1 = \mathbf{e}_2,</math>

:<math>\mathbf{e}_2 \times \mathbf{e}_3 = \mathbf{e}_5, \quad \mathbf{e}_3 \times \mathbf{e}_5 = \mathbf{e}_2, \quad \mathbf{e}_5 \times \mathbf{e}_2 = \mathbf{e}_3,</math>

:<math>\mathbf{e}_3 \times \mathbf{e}_4 = \mathbf{e}_6, \quad \mathbf{e}_4 \times \mathbf{e}_6 = \mathbf{e}_3, \quad \mathbf{e}_6 \times \mathbf{e}_3 = \mathbf{e}_4,</math>

:<math>\mathbf{e}_4 \times \mathbf{e}_5 = \mathbf{e}_7, \quad \mathbf{e}_5 \times \mathbf{e}_7 = \mathbf{e}_4, \quad \mathbf{e}_7 \times \mathbf{e}_4 = \mathbf{e}_5,</math>

:<math>\mathbf{e}_5 \times \mathbf{e}_6 = \mathbf{e}_1, \quad \mathbf{e}_6 \times \mathbf{e}_1 = \mathbf{e}_5, \quad \mathbf{e}_1 \times \mathbf{e}_5 = \mathbf{e}_6,</math>

:<math>\mathbf{e}_6 \times \mathbf{e}_7 = \mathbf{e}_2, \quad \mathbf{e}_7 \times \mathbf{e}_2 = \mathbf{e}_6, \quad \mathbf{e}_2 \times \mathbf{e}_6 = \mathbf{e}_7,</math>

:<math>\mathbf{e}_7 \times \mathbf{e}_1 = \mathbf{e}_3, \quad \mathbf{e}_1 \times \mathbf{e}_3 = \mathbf{e}_7, \quad \mathbf{e}_3 \times \mathbf{e}_7 = \mathbf{e}_1.</math>

More compactly this rule can be written as

: <math>\mathbf{e}_i \times \mathbf{e}_{i+1} = \mathbf{e}_{i+3}</math>

with ''i'' = 1...7 [[Modular arithmetic|modulo]] 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,
: <math>\mathbf{e}_i \times \left( \mathbf{e}_i \times \mathbf{e}_{i+1}\right) =-\mathbf{e}_{i+1} = \mathbf{e}_i \times \mathbf{e}_{i+3} \ ,</math>
which produces diagonals further out, and so on.

The '''e'''<sub>j</sub> component of cross product '''x''' × '''y''' is given by selecting all occurrences of '''e'''<sub>j</sub> in the table and collecting the corresponding components of '''x''' from the left column and of '''y''' from the top row. The result is:

:<math>\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}</math>

As the cross product is bilinear the operator '''x'''×– can be written as a matrix, which takes the form{{Citation needed|date=July 2010}}

:<math>T_{\mathbf x} = \begin{bmatrix}
 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{bmatrix}.</math>

The cross product is then given by

:<math>\mathbf{x} \times \mathbf{y} = T_{\mathbf{x}} \mathbf{y}.</math>

===Different multiplication tables===

[[File:Fano plane for 7-D cross product.svg|thumb|Fano planes for the two multiplication tables used here.]]
Two different multiplication tables have been used in this article, and there are more.<ref name=Parra/><ref name=T_Smith>
Further discussion of the tables and the connection of the Fano plane to these tables is found here: {{cite web |url=http://www.tony5m17h.net/480op.html |title=Octonion products and lattices |author=Tony Smith |access-date=2018-05-12}}
</ref> These multiplication tables are characterized by the [[Fano plane]],<ref name=Fauser>

{{cite book |title=Clifford Algebras and Their Applications in Mathematical Physics: Algebra and physics |author1=Rafał Abłamowicz |author2=Bertfried Fauser |page=26 |url=https://books.google.com/books?id=yvCC94xzJG8C&pg=PA26 |isbn=0-8176-4182-3 |year=2000 |publisher=Springer }}

</ref><ref name=Manogue/> 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'''<sub>''i''</sub> × '''e'''<sub>''j''</sub> = '''e'''<sub>''k''</sub>. 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 '''e'''<sub>1</sub> in the above listing is obtained by following the three paths connected to '''e'''<sub>1</sub> in the lower Fano diagram: the circular path '''e'''<sub>2</sub> × '''e'''<sub>4</sub>, the diagonal path '''e'''<sub>3</sub> × '''e'''<sub>7</sub>, and the edge path '''e'''<sub>6</sub> × '''e'''<sub>1</sub> = '''e'''<sub>5</sub> rearranged using one of the above identities as:

:<math>\mathbf{e}_6 \times \left( \mathbf{e}_6 \times \mathbf{e}_1 \right) = -\mathbf{e}_1 = \mathbf{e}_6 \times \mathbf{e}_5 , </math>

or

:<math> \mathbf{e}_5 \times \mathbf{e}_6 =\mathbf{e}_1 , </math>
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.<ref name=Manogue>
{{cite journal |title=Octonionic representations of Clifford algebras and triality |author1=Jörg Schray |author2=Corinne A. Manogue |journal=Foundations of Physics |pages=17–70 |volume=26 |year=1996 |issue= 1/January |doi=10.1007/BF02058887 |arxiv=hep-th/9407179 |bibcode=1996FoPh...26...17S |s2cid=119604596 }} Available as [https://arxiv.org/abs/hep-th/9407179v1 ArXive preprint] Figure 1 is located [https://arxiv.org/PS_cache/hep-th/ps/9407/9407179v1.fig1-1.png here].
</ref>

===Using geometric algebra===
The product can also be calculated using [[Geometric algebra|geometric algebra]]. The product starts with the exterior product, a [[Bivector|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, 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]] 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

:<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 [[Geometric algebra#Extensions of the inner and outer products|left contraction]] operator from geometric algebra.<ref name=Lounesto/><ref name= "Abłamowicz0">

{{cite book |title=Clifford algebras: applications to mathematics, physics, and engineering |author=Bertfried Fauser |editor1=Pertti Lounesto |editor2=Rafał Abłamowicz |chapter-url=https://books.google.com/books?id=b6mbSCv_MHMC&pg=PA292 |chapter=§18.4.2 Contractions |publisher=Birkhäuser |year=2004 |pages=292 ''ff'' |isbn=0-8176-3525-4}}

</ref>

==Relation to the octonions==
Just as the 3-dimensional cross product can be expressed in terms of the [[Quaternion|quaternion]]s, the 7-dimensional cross product can be expressed in terms of the [[Octonion|octonion]]s. After identifying <math>\mathbb{R}^7</math> with the imaginary octonions (the [[Orthogonal complement|orthogonal complement]] of the real line in <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 ''V'' 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}\cdot\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.<ref name=Baez>
{{cite journal
 |title       = The Octonions
 |author      = John C. Baez
 |url         = http://math.ucr.edu/home/baez/octonions/oct.pdf
 |year        = 2002
 |journal     = Bull. Amer. Math. Soc.
 |volume      = 39
 |issue = 2
 |pages = 145–205
 |doi         = 10.1090/s0273-0979-01-00934-x
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20100707194509/http://math.ucr.edu/home/baez/octonions/oct.pdf
 |archive-date = 2010-07-07
|arxiv= math/0105155
 |s2cid = 586512
 }}
</ref>

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 (normed division algebras)|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.<ref>

{{cite journal | first = Alberto | last = Elduque | title = Vector cross products | year = 2004 | url = http://www.unizar.es/matematicas/algebra/elduque/Talks/crossproducts.pdf}}

</ref><ref>

{{cite journal | first = Erik | last = Darpö | title = Vector product algebras |journal=Bulletin of the London Mathematical Society |volume= 41 |pages=898–902| year = 2009 | doi =10.1112/blms/bdp066 |issue=5|arxiv=0810.5464 | s2cid = 122615967 }} See also: {{cite journal | citeseerx = 10.1.1.66.4 | title = Real vector product algebras }}</ref>

The failure of the 7-dimension cross product to satisfy the Jacobi identity is due 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|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 {{nowrap|'''x''' × '''y'''}} 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, [[Orthogonal group|SO(7)]]. Instead it is invariant under the exceptional Lie group [[G2 (mathematics)|G<sub>2</sub>]], a subgroup of SO(7).<ref name=Lounesto/><ref name =Baez/>

==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.<ref name=k_vectors>Lounesto, §7.5: ''Cross products of k vectors in <math>\mathbb{R}^n</math>'', p. 98</ref><ref name=Gallier>

{{cite book |title=Geometric methods and applications: for computer science and engineering |author=Jean H. Gallier |chapter-url=https://books.google.com/books?id=CTHaW9ft1ZMC&pg=PA244 |page=[https://archive.org/details/geometricmethods0000gall/page/244 244] |chapter=Problem 7.10 (2) |isbn=0-387-95044-3 |year=2001 |publisher=Springer |url=https://archive.org/details/geometricmethods0000gall/page/244 }}
</ref> 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 {{nowrap|''n'' − 1}} vectors can be used. The magnitude of the product should equal the volume of the [[Parallelepiped#Parallelotope|parallelotope]] with the vectors as edges, which can be calculated using the [[Gramian matrix#Gram determinant|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 <math>i = 1,\ \dots\ , k</math>.
*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 ''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
<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 {{nowrap|1='''w''' = −'''ve'''<sub>12...7</sub>}} 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 <math>V^d \to V</math> (where <math>V</math> is <math>\mathbb{R}^n</math> endowed with the Euclidean inner product and <math> d \geq 2 </math>) 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 <math>n = 3, d = 2</math>, (II) for <math>n = 7, d = 3</math>, (III) for <math>n = 8, d = 3</math>, and (IV) for any <math> d = n - 1 </math>.<ref name=Massey0/>

==See also==
* [[Composition algebra]]

==Notes==
{{Reflist}}

==References==
*{{cite journal | last1 = Brown | first1 = Robert B. | last2 = Gray | first2= Alfred | year = 1967 | title = Vector cross products | journal = [[Commentarii Mathematici Helvetici]] | volume = 42 | issue = 1 | pages = 222–236 | doi = 10.1007/BF02564418 | s2cid = 121135913 }}
*{{cite book | last = Lounesto | first = Pertti | title = Clifford algebras and spinors | publisher = Cambridge University Press | location = Cambridge, UK | year = 2001 | isbn=0-521-00551-5 | url =https://books.google.com/books?id=kOsybQWDK4oC}}
*{{cite journal | first = Z.K. | last = Silagadze | title = Multi-dimensional vector product |year = 2002 |journal=J Phys A |volume=35 |pages=4949–4953 |doi=10.1088/0305-4470/35/23/310 | issue = 23|arxiv=math/0204357 |bibcode=2002JPhA...35.4949S | s2cid = 119165783 }} Also available as ArXiv reprint {{arxiv|math.RA/0204357}}.
*{{cite journal | first = W.S. | last = Massey | title = Cross products of vectors in higher dimensional Euclidean spaces | year = 1983 | jstor = 2323537| pages = 697–701 | volume = 90 | issue = 10 | journal = The American Mathematical Monthly | doi = 10.2307/2323537}}

{{Linear algebra}}

{{DEFAULTSORT:Seven-Dimensional Cross Product}}
[[Category:Bilinear operators]]

{{Sourceattribution|Seven-dimensional cross product|1}}