- Seven-dimensional cross product
In

mathematics , the**seven-dimensional cross product**is abinary operation on vectors in a seven-dimensionalEuclidean space . It is a generalization of the ordinary three-dimensionalcross product . The seven-dimensional cross product has the same relationship to theoctonion s as the three-dimensional cross product does to thequaternion s.Nontrivial binary cross products exist only in 3 and 7 dimensions. There are no higher-dimensional analogs.

**Characteristic properties**A cross product in an "n"-dimensional

Euclidean space "V" is defined as abilinear map :"V" × "V" → "V"such that

***x**· (**x**×**y**) =**y**· (**x**×**y**) = 0, and

*|**x**×**y**|^{2}= |**x**|^{2}|**y**|^{2}− (**x**·**y**)^{2}for all**x**and**y**in "V". Here**x**·**y**denotes the standard Euclideandot product . The first of these properties states that the cross product should be perpendicular to each of its arguments. The second states that the norm of the cross product should be equal to the area of theparallelogram formed by the arguments. This is equivalent to the statement

*|**x**×**y**| = |**x**||**y**|sin θwhere θ is the inner angle between**x**and**y**.It has been shown that nontrivial cross products exist only for "n" = 3 and "n" = 7. The reason has to do with the connection between cross products and

normed division algebra s (see the section below).Some properties that follow from the above characterization, and are therefore true in seven dimensions, are:

***x**×**y**= −**y**×**x**

***x**· (**y**×**z**) =**y**· (**z**×**x**) =**z**· (**x**×**y**)

***x**× (**x**×**y**) = −|**x**|^{2}**y**+ (**x**·**y**)**x**.Some properties of the 3-dimensional cross product which do not generally hold for the 7-dimensional one are the

vector triple product formula (equality holds in three dimensions):**x**× (**y**×**z**) ≠ (**x**·**z**)**y**− (**x**·**y**)**z**and theJacobi identity ::**x**× (**y**×**z**) +**y**× (**z**×**x**) +**z**× (**x**×**y**) ≠**0**.Both of these fail due to the nonassociativity of the octonions.**Coordinate expression**Unlike the 3-dimensional cross product, the 7-dimension cross product is not unique (up to a sign). This is because there is more than one direction perpendicular to any given plane. However, any two cross products differ by an orthogonal transformation.

One possible cross product on

**R**^{7}is given by the rules

***e**_{1}×**e**_{2}=**e**_{4}

***e**_{"i"}×**e**_{"j"}=**e**_{"k"}implies that

****e**_{2"i"}×**e**_{2"j"}=**e**_{2"k"}

****e**_{"i"+1}×**e**_{"j"+1}=**e**_{"k"+1}where {**e**_{"i"}} is thestandard basis for**R**^{7}and the indices are read from 1 to 7 modulo 7. Together with the fact that the cross product must be antisymmetric these rules complete determine the cross product.Explicitly, the cross product is given by the expression:$egin\{align\}mathbf\{x\}\; imesmathbf\{y\}\; =\; (x\_2y\_4\; -\; x\_4y\_2\; +\; x\_3y\_7\; -\; x\_7y\_3\; +\; x\_5y\_6\; -\; x\_6y\_5),mathbf\{e\}\_1\; \backslash \; \{\}+\; (x\_3y\_5\; -\; x\_5y\_3\; +\; x\_4y\_1\; -\; x\_1y\_4\; +\; x\_6y\_7\; -\; x\_7y\_6),mathbf\{e\}\_2\; \backslash \; \{\}+\; (x\_4y\_6\; -\; x\_6y\_4\; +\; x\_5y\_2\; -\; x\_2y\_5\; +\; x\_7y\_1\; -\; x\_1y\_7),mathbf\{e\}\_3\; \backslash \; \{\}+\; (x\_5y\_7\; -\; x\_7y\_5\; +\; x\_6y\_3\; -\; x\_3y\_6\; +\; x\_1y\_2\; -\; x\_2y\_1),mathbf\{e\}\_4\; \backslash \; \{\}+\; (x\_6y\_1\; -\; x\_1y\_6\; +\; x\_7y\_4\; -\; x\_4y\_7\; +\; x\_2y\_3\; -\; x\_3y\_2),mathbf\{e\}\_5\; \backslash \; \{\}+\; (x\_7y\_2\; -\; x\_2y\_7\; +\; x\_1y\_5\; -\; x\_5y\_1\; +\; x\_3y\_4\; -\; x\_4y\_3),mathbf\{e\}\_6\; \backslash \; \{\}+\; (x\_1y\_3\; -\; x\_3y\_1\; +\; x\_2y\_6\; -\; x\_6y\_2\; +\; x\_4y\_5\; -\; x\_5y\_4),mathbf\{e\}\_7.\; \backslash end\{align\}$

The 7-dimensional cross product can also be written as a sum of seven 3-dimensional cross products. Let π

_{"i"}denote the orthogonal projection of**R**^{7}onto the 3-dimensional subspace spanned by**e**_{"i"+1},**e**_{"i"+2}, and**e**_{"i"+4}. Then:$mathbf\{x\}\; imesmathbf\{y\}\; =\; sum\_\{i=1\}^\{7\}pi\_i(mathbf\; x)\; imespi\_i(mathbf\; y).$Since the cross product is bilinear, one can write the operator

**x**×– as a matrix. This matrix has the form:$T\_\{mathbf\; x\}\; =\; left\; [egin\{matrix\}\; 0\; -x\_4\; -x\_7\; x\_2\; -x\_6\; x\_5\; x\_3\; \backslash \; x\_4\; 0\; -x\_5\; -x\_1\; x\_3\; -x\_7\; x\_6\; \backslash \; x\_7\; x\_5\; 0\; -x\_6\; -x\_2\; x\_4\; -x\_1\; \backslash -x\_2\; x\_1\; x\_6\; 0\; -x\_7\; -x\_3\; x\_5\; \backslash \; x\_6\; -x\_3\; x\_2\; x\_7\; 0\; -x\_1\; -x\_4\; \backslash -x\_5\; x\_7\; -x\_4\; x\_3\; x\_1\; 0\; -x\_2\; \backslash -x\_3\; -x\_6\; x\_1\; -x\_5\; x\_4\; x\_2\; 0end\{matrix\}\; ight]\; .$The cross product**x**×**y**is then given by "T"_{x}(**y**).**Relation to the octonions**Just as the 3-dimensional cross product can be expressed in terms of the

quaternion s, the 7-dimensional cross product can be expressed in terms of theoctonion s. After identifying**R**^{7}with the imaginary octonions (theorthogonal complement of the identity in**O**), the cross product is given in terms of octonion multiplication by:$mathbf\; x\; imes\; mathbf\; y\; =\; mathrm\{Im\}(mathbf\{xy\})\; =\; frac\{1\}\{2\}(mathbf\{xy\}-mathbf\{yx\}).$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::$(a,mathbf\{x\})(b,mathbf\{y\})\; =\; (ab\; -\; mathbf\{x\}cdotmathbf\{y\},\; amathbf\; y\; +\; bmathbf\; x\; +\; mathbf\{x\}\; imesmathbf\{y\}).$The space**R**⊕"V" with this multiplication is then isomorphic to the octonions.The cross product only exists in dimensions 3 and 7 since 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 . Such algebras only exist in dimensions 1, 2, 4, and 8. (There is, of course, a trivial cross product is dimension 1 coming from thecomplex number s; trivial since the complex numbers arecommutative ).The failure of the 7-dimension cross product to satisfy the Jacobi identity is due to the nonassociativity of the octonions. In fact, :$mathbf\{x\}\; imes(mathbf\{y\}\; imesmathbf\{z\})\; +\; mathbf\{y\}\; imes(mathbf\{z\}\; imesmathbf\{x\})\; +\; mathbf\{z\}\; imes(mathbf\{x\}\; imesmathbf\{y\})\; =\; -frac\{3\}\{2\}\; [mathbf\; x,\; mathbf\; y,\; mathbf\; z]\; .$where [

**x**,**y**,**z**] is theassociator .**References***cite journal | last = Brown | first = Robert B. | coauthors = Gray, Alfred | year = 1967 | title = Vector cross products | journal = Commentarii Mathematici Helvetici | volume = 42 | issue = 1 | pages = 222–236 | doi = 10.1007/BF02564418

*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

*cite paper | first = Z.K. | last = Silagadze | title = Multi-dimensional vector product | date = 2002 | id = arxiv|archive=math.RA|id=0204357

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