Seven-dimensional cross product

In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in a seven-dimensional space. It assigns to any two vectors a, b in ℝ⁷ a vector a × b ∈ ℝ⁷.^[1] In seven dimensions there also exists a cross product involving six vectors (that is, linear but not binary), discussed briefly in the section on generalizations.

As in the more familiar three-dimensional cross product, 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. The seven dimensional cross product has the same relationship to octonions as the three-dimensional cross product does to quaternions, 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.^[2]

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.^[3]^[4] 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.^[5] 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..

Alternate indexing schemes
$Number$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$Letter$	$i$	$j$	$k$	$l$	$il$	$jl$	$kl$
$Alternate$	$i$	$j$	$k$	$l$	$m$	$n$	$o$

Cayley's sample multiplication table
×	$e 1$	$e 2$	$e 3$	$e 4$	$e 5$	$e 6$	$e 7$
$e 1$	$0$	$e 3$	$- e 2$	$e 5$	$- e 4$	$- e 7$	$e 6$
$e 2$	$- e 3$	$0$	$e 1$	$e 6$	$e 7$	$- e 4$	$- e 5$
$e 3$	$e 2$	$- e 1$	$0$	$e 7$	$- e 6$	$e 5$	$- e 4$
$e 4$	$- e 5$	$- e 6$	$- e 7$	$0$	$e 1$	$e 2$	$e 3$
$e 5$	$e 4$	$- e 7$	$e 6$	$- e 1$	$0$	$- e 3$	$e 2$
$e 6$	$e 7$	$e 4$	$- e 5$	$- e 2$	$e 3$	$0$	$- e 1$
$e 7$	$- e 6$	$e 5$	$e 4$	$- e 3$	$- e 2$	$e 1$	$0$

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^[4]

\mathbf {e} _{i}\mathbf {\times } \mathbf {e} _{j}=\varepsilon _{ijk}\mathbf {e} _{k}\ ,

where $\varepsilon _{ijk}$ 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

\left(\mathbf {x\times y} \right)_{1}=x_{2}y_{3}-x_{3}y_{2}+x_{4}y_{5}-x_{5}y_{4}+x_{7}y_{6}-x_{6}y_{7}=-\left(\mathbf {y\times x} \right)_{1}\ .

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 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^[1]^[6]

\mathbf {x} \cdot (\mathbf {x} \times \mathbf {y} )=(\mathbf {x} \times \mathbf {y} )\mathbf {\cdot y} =0

(orthogonality),

and:

|\mathbf {x} \times \mathbf {y} |^{2}=|\mathbf {x} |^{2}|\mathbf {y} |^{2}-(\mathbf {x} \cdot \mathbf {y} )^{2}

(magnitude),

where (x·y) is the Euclidean dot product and |x| is the vector norm. 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 θ between the vectors^[7] is^[8]

|\mathbf {x} \times \mathbf {y} |=|\mathbf {x} ||\mathbf {y} |\sin \theta ,

or the area of the parallelogram in the plane of x and y with the two vectors as sides.^[9] As a third alternative the following can be shown to be equivalent to either expression for the magnitude:^[10]

|\mathbf {x} \times \mathbf {y} |=|\mathbf {x} ||\mathbf {y} |~{\mbox{if}}\ \left(\mathbf {x} \cdot \mathbf {y} \right)=0.

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.^[8]^[2]^[10] 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 algebras 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.^[11]

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:^[8]

“…for the cross product x × y in ℝ⁷ there are also other planes than the linear span of x and y giving the same direction as x × y”
— Pertti Lounesto, Clifford algebras and spinors, p. 97

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:

\mathbf {x} \times \mathbf {y} =-\mathbf {y} \times \mathbf {x}

(anticommutativity),

\mathbf {x} \cdot (\mathbf {y} \times \mathbf {z} )=\mathbf {y} \cdot (\mathbf {z} \times \mathbf {x} )=\mathbf {z} \cdot (\mathbf {x} \times \mathbf {y} )

(scalar triple product),

(\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}

(Malcev identity),^[8]

\mathbf {x} \times (\mathbf {x} \times \mathbf {y} )=-|\mathbf {x} |^{2}\mathbf {y} +(\mathbf {x} \cdot \mathbf {y} )\mathbf {x} .

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

\mathbf {x} \times (\mathbf {y} \times \mathbf {z} )=(\mathbf {x} \cdot \mathbf {z} )\mathbf {y} -(\mathbf {x} \cdot \mathbf {y} )\mathbf {z}

(vector triple product),

\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

(Jacobi identity).^[8]

Coordinate expressions

To define a particular cross product, an orthonormal basis {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.^[5] 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.

Lounesto's multiplication table
×	$e 1$	$e 2$	$e 3$	$e 4$	$e 5$	$e 6$	$e 7$
$e 1$	$0$	$e 4$	$e 7$	$- e 2$	$e 6$	$- e 5$	$- e 3$
$e 2$	$- e 4$	$0$	$e 5$	$e 1$	$- e 3$	$e 7$	$- e 6$
$e 3$	$- e 7$	$- e 5$	$0$	$e 6$	$e 2$	$- e 4$	$e 1$
$e 4$	$e 2$	$- e 1$	$- e 6$	$0$	$e 7$	$e 3$	$- e 5$
$e 5$	$- e 6$	$e 3$	$- e 2$	$- e 7$	$0$	$e 1$	$e 4$
$e 6$	$e 5$	$- e 7$	$e 4$	$- e 3$	$- e 1$	$0$	$e 2$
$e 7$	$e 3$	$e 6$	$- e 1$	$e 5$	$- e 4$	$- e 2$	$0$

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^[8]

\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},

\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},

\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},

\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},

\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},

\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},

\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}.

More compactly this rule can be written as

\mathbf {e} _{i}\times \mathbf {e} _{i+1}=\mathbf {e} _{i+3}

with i = 1...7 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,

\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}\ ,

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:

{\begin{aligned}\mathbf {x} \times \mathbf {y} =(x_{2}y_{4}-x_{4}y_{2}+x_{3}y_{7}-x_{7}y_{3}+x_{5}y_{6}-x_{6}y_{5})\,&\mathbf {e} _{1}\\{}+(x_{3}y_{5}-x_{5}y_{3}+x_{4}y_{1}-x_{1}y_{4}+x_{6}y_{7}-x_{7}y_{6})\,&\mathbf {e} _{2}\\{}+(x_{4}y_{6}-x_{6}y_{4}+x_{5}y_{2}-x_{2}y_{5}+x_{7}y_{1}-x_{1}y_{7})\,&\mathbf {e} _{3}\\{}+(x_{5}y_{7}-x_{7}y_{5}+x_{6}y_{3}-x_{3}y_{6}+x_{1}y_{2}-x_{2}y_{1})\,&\mathbf {e} _{4}\\{}+(x_{6}y_{1}-x_{1}y_{6}+x_{7}y_{4}-x_{4}y_{7}+x_{2}y_{3}-x_{3}y_{2})\,&\mathbf {e} _{5}\\{}+(x_{7}y_{2}-x_{2}y_{7}+x_{1}y_{5}-x_{5}y_{1}+x_{3}y_{4}-x_{4}y_{3})\,&\mathbf {e} _{6}\\{}+(x_{1}y_{3}-x_{3}y_{1}+x_{2}y_{6}-x_{6}y_{2}+x_{4}y_{5}-x_{5}y_{4})\,&\mathbf {e} _{7}.\\\end{aligned}}

As the cross product is bilinear the operator x×– can be written as a matrix, which takes the form^{[citation needed]}

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}}.

The cross product is then given by

\mathbf {x} \times \mathbf {y} =T_{\mathbf {x} }(\mathbf {y} ).

Different multiplication tables

Two different multiplication tables have been used in this article, and there are more.^[5]^[12] These multiplication tables are characterized by the Fano plane,^[13]^[14] 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 {e_{6}\times } \left(\mathbf {e_{6}\times e_{1}} \right)=-\mathbf {e_{1}} =\mathbf {e_{6}\times e_{5}} \ ,

or

\mathbf {e_{5}\times e_{6}} =\mathbf {e_{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. The question arises: how many multiplication tables are there?^[14]

The question of possible multiplication tables arises, for example, when one reads another article on octonions, which uses a different one from the one given by [Cayley, say]. Usually it is remarked that all 480 possible ones are equivalent, that is, given an octonionic algebra with a multiplication table and any other valid multiplication table, one can choose a basis such that the multiplication follows the new table in this basis. One may also take the point of view, that there exist different octonionic algebras, that is, algebras with different multiplication tables. With this interpretation...all these octonionic algebras are isomorphic.
— Jörg Schray, Corinne A Manogue, Octonionic representations of Clifford algebras and triality (1994)

Using geometric algebra

The product can also be calculated using geometric algebra. The product starts with the exterior product, a bivector valued product of two vectors:

\mathbf {B} =\mathbf {x} \wedge \mathbf {y} ={\frac {1}{2}}(\mathbf {xy} -\mathbf {yx} ).

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 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 {v} =\mathbf {e} _{124}+\mathbf {e} _{235}+\mathbf {e} _{346}+\mathbf {e} _{457}+\mathbf {e} _{561}+\mathbf {e} _{672}+\mathbf {e} _{713}.

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

\mathbf {x} \times \mathbf {y} =-(\mathbf {x} \wedge \mathbf {y} )\lrcorner \mathbf {v}

where ⌋ is the left contraction operator from geometric algebra.^[8]^[15]

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 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

\mathbf {x} \times \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} \cdot \mathbf {y} ,a\mathbf {y} +b\mathbf {x} +\mathbf {x} \times \mathbf {y} ).

The space R⊕V with this multiplication is then isomorphic to the octonions.^[16]

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.^[17]^[18]

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} \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} ]

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

Rotations

In three dimensions the cross product is invariant under the group of the rotation group, SO(3), so the cross product of x and y after they are rotated is the image of 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, SO(7). Instead it is invariant under the exceptional Lie group G₂, a subgroup of SO(7).^[16]^[8]

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.^[19]^[20] 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

\left(\mathbf {a_{1}\times \ \cdots \ \times a_{k}} \right)\mathbf {\cdot a_{j}} =0

(orthogonality)

|\mathbf {a_{1}\times \ \cdots \ \times a_{k}} |^{2}=\det(\mathbf {a_{i}\cdot a_{j}} )={\begin{vmatrix}\mathbf {a_{1}\cdot a_{1}} &\mathbf {a_{1}\cdot a_{2}} &\dots &\mathbf {a_{1}\cdot a_{k}} \\\mathbf {a_{2}\cdot a_{1}} &\mathbf {a_{2}\cdot a_{2}} &\dots &\mathbf {a_{2}\cdot a_{k}} \\\dots &\dots &\dots &\dots \\\mathbf {a_{k}\cdot a_{1}} &\mathbf {a_{k}\cdot a_{2}} &\dots &\mathbf {a_{k}\cdot a_{k}} \\\end{vmatrix}}

(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

|\mathbf {x} \times \mathbf {y} |^{2}={\begin{vmatrix}\mathbf {x\cdot x} &\mathbf {x\cdot y} \\\mathbf {y\cdot x} &\mathbf {y\cdot y} \\\end{vmatrix}}=|\mathbf {x} |^{2}|\mathbf {y} |^{2}-(\mathbf {x} \cdot \mathbf {y} )^{2}

,

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

\mathbf {a} \times \mathbf {b} \times \mathbf {c} =(\mathbf {a} \wedge \mathbf {b} \wedge \mathbf {c} )\lrcorner (\mathbf {w} -\mathbf {ve} _{8})

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.

Notes

^ ^a ^b WS Massey (1993). "Cross products of vectors in higher dimensional Euclidean spaces". The American Mathematical Monthly. 90 (10). Mathematical Association of America: 697–701. doi:10.2307/2323537. {{cite journal}}: Invalid |ref=harv (help)
^ ^a ^b WS Massey (1993). "cited work". The American Mathematical Monthly. 90 (10): 697–701. 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. {{cite journal}}: Invalid |ref=harv (help)
^ This table is due to Arthur Cayley (1845) and John T. Graves (1843). See G Gentili, C Stoppato, DC Struppa and F Vlacci (2009). "Recent developments for regular functions of a hypercomplex variable". In Irene Sabadini, M Shapiro, F Sommen (ed.). Hypercomplex analysis (Conference on quaternionic and Clifford analysis; proceedings ed.). Birkaüser. p. 168. ISBN 9783764398927.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ ^a ^b Lev Vasilʹevitch Sabinin, Larissa Sbitneva, I. P. Shestakov (2006). "§17.2 Octonion algebra and its regular bimodule representation". Non-associative algebra and its applications. CRC Press. p. 235. ISBN 0824726693. {{cite book}}: Invalid |ref=harv (help)CS1 maint: multiple names: authors list (link)
^ ^a ^b ^c Rafał Abłamowicz, Pertti Lounesto, Josep M. Parra (1996). "§ Four ocotonionic basis numberings". Clifford algebras with numeric and symbolic computations. Birkhäuser. p. 202. ISBN 0817639071.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Mappings are restricted to be bilinear by (Massey 1993) and Robert B Brown and Alfred Gray (1967). "Vector cross products". Commentarii Mathematici Helvetici. 42 (Number 1/December). Birkhäuser Basel: 222–236. doi:10.1007/BF02564418. {{cite journal}}: |issue= has extra text (help).
^ The definition of angle in n-dimensions ordinarily is defined in terms of the dot product as:
$(\mathbf {x\cdot y} )=|\mathbf {x} ||\mathbf {y} |\ \cos \theta \ ,\ \mathrm {in\ the\ range} \ (-\pi <\theta \leq \pi )\ ,$
where θ is the angle between the vectors. Consequently, this property of the cross product provides its magnitude as:
$|\mathbf {x\times y} |^{2}=|\mathbf {x} |^{2}|\mathbf {y} |^{2}\left(1-\cos ^{2}\theta \right)\ .$
From the Pythagorean trigonometric identity this magnitude equals
$|\mathbf {x} \times \mathbf {y} |=|\mathbf {x} ||\mathbf {y} |\sin \theta$ .
See Francis Begnaud Hildebrand (1992). Methods of applied mathematics (Reprint of Prentice-Hall 1965 2nd ed.). Courier Dover Publications. p. 24. ISBN 0486670023.
^ ^a ^b ^c ^d ^e ^f ^g ^h Lounesto, pp. 96-97
^ Kendall, M. G. (2004). A Course in the Geometry of N Dimensions. Courier Dover Publications. p. 19. ISBN 0486439275.
^ ^a ^b Z.K. Silagadze (2002). "Multi-dimensional vector product". arXiv:math.RA/0204357. {{cite journal}}: Cite journal requires |journal= (help)
^ Nathan Jacobson (2009). Basic algebra I (Reprint of Freeman 1974 2nd ed.). Dover Publications. pp. 417–427. ISBN 0486471896.
^ Further discussion of the tables and the connection of the Fano plane to these tables is found here: Tony Smith. "Octonion products and lattices". Retrieved 2010-07-11.
^ Rafał Abłamowicz, Bertfried Fauser (2000). Clifford Algebras and Their Applications in Mathematical Physics: Algebra and physics. Springer. p. 26. ISBN 0817641823.
^ ^a ^b Jörg Schray, Corinne A. Manogue (1996). "Octonionic representations of Clifford algebras and triality". Foundations of physics. 26 (1/January). Springer: 17–70. doi:10.1007/BF02058887. {{cite journal}}: Invalid |ref=harv (help) Available as ArXive preprint Figure 1 is located here.
^ Bertfried Fauser (2004). "§18.4.2 Contractions". In Pertti Lounesto, Rafał Abłamowicz (ed.). Clifford algebras: applications to mathematics, physics, and engineering. Birkhäuser. pp. 292 ff. ISBN 0817635254.
^ ^a ^b John C. Baez (2001). "The Octonions" (PDF). Bull. Amer. Math. 39: 38. {{cite journal}}: Invalid |ref=harv (help)
^ Elduque, Alberto (2004). "Vector cross products" (PDF). {{cite journal}}: Cite journal requires |journal= (help)
^ Darpö, Erik (2009). "Vector product algebras". Bulletin of the London Mathematical Society. Vol. 41 (5): 898–902. doi:10.1112/blms/bdp066. {{cite journal}}: |volume= has extra text (help) See also Real vector product algebras
^ Lounesto, §7.5: Cross products of k vectors in ℝⁿ, p. 98
^ Jean H. Gallier (2001). "Problem 7.10 (2)". Geometric methods and applications: for computer science and engineering. Springer. p. 244. ISBN 0387950443.

References

Brown, Robert B. (1967). "Vector cross products". Commentarii Mathematici Helvetici. 42 (1): 222–236. doi:10.1007/BF02564418. {{cite journal}}: Invalid |ref=harv (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
Lounesto, Pertti (2001). Clifford algebras and spinors. Cambridge, UK: Cambridge University Press. ISBN 0-521-00551-5. {{cite book}}: Invalid |ref=harv (help)
Silagadze, Z.K. (2002). "Multi-dimensional vector product". J Phys A: Math Gen. 35: 4949. doi:10.1088/0305-4470/35/23/310. Also available as ArXiv reprint arXiv:math.RA/0204357.
Massey, W.S. (1993). "Cross products of vectors in higher dimensional Euclidian spaces". JSTOR 2323537. {{cite journal}}: Cite journal requires |journal= (help); Invalid |ref=harv (help)

[Massey0-1] WS Massey (1993). "Cross products of vectors in higher dimensional Euclidean spaces". The American Mathematical Monthly. 90 (10). Mathematical Association of America: 697–701. doi:10.2307/2323537. {{cite journal}}: Invalid |ref=harv (help)

[Massey2-2] WS Massey (1993). "cited work". The American Mathematical Monthly. 90 (10): 697–701. 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. {{cite journal}}: Invalid |ref=harv (help)

[Cayley-3] This table is due to Arthur Cayley (1845) and John T. Graves (1843). See G Gentili, C Stoppato, DC Struppa and F Vlacci (2009). "Recent developments for regular functions of a hypercomplex variable". In Irene Sabadini, M Shapiro, F Sommen (ed.). Hypercomplex analysis (Conference on quaternionic and Clifford analysis; proceedings ed.). Birkaüser. p. 168. ISBN 9783764398927.{{cite book}}: CS1 maint: multiple names: authors list (link)

[Shestakov-4] Lev Vasilʹevitch Sabinin, Larissa Sbitneva, I. P. Shestakov (2006). "§17.2 Octonion algebra and its regular bimodule representation". Non-associative algebra and its applications. CRC Press. p. 235. ISBN 0824726693. {{cite book}}: Invalid |ref=harv (help)CS1 maint: multiple names: authors list (link)

[Parra-5] Rafał Abłamowicz, Pertti Lounesto, Josep M. Parra (1996). "§ Four ocotonionic basis numberings". Clifford algebras with numeric and symbolic computations. Birkhäuser. p. 202. ISBN 0817639071.{{cite book}}: CS1 maint: multiple names: authors list (link)

[Brown-6] Mappings are restricted to be bilinear by (Massey 1993) and Robert B Brown and Alfred Gray (1967). "Vector cross products". Commentarii Mathematici Helvetici. 42 (Number 1/December). Birkhäuser Basel: 222–236. doi:10.1007/BF02564418. {{cite journal}}: |issue= has extra text (help).

[Hildebrand-7] The definition of angle in n-dimensions ordinarily is defined in terms of the dot product as:
$(\mathbf {x\cdot y} )=|\mathbf {x} ||\mathbf {y} |\ \cos \theta \ ,\ \mathrm {in\ the\ range} \ (-\pi <\theta \leq \pi )\ ,$
where θ is the angle between the vectors. Consequently, this property of the cross product provides its magnitude as:
$|\mathbf {x\times y} |^{2}=|\mathbf {x} |^{2}|\mathbf {y} |^{2}\left(1-\cos ^{2}\theta \right)\ .$
From the Pythagorean trigonometric identity this magnitude equals
$|\mathbf {x} \times \mathbf {y} |=|\mathbf {x} ||\mathbf {y} |\sin \theta$ .
See Francis Begnaud Hildebrand (1992). Methods of applied mathematics (Reprint of Prentice-Hall 1965 2nd ed.). Courier Dover Publications. p. 24. ISBN 0486670023.

[Lounesto-8] ^ ^a ^b ^c ^d ^e ^f ^g ^h Lounesto, pp. 96-97

[9] Kendall, M. G. (2004). A Course in the Geometry of N Dimensions. Courier Dover Publications. p. 19. ISBN 0486439275.

[Silagadze1-10] Z.K. Silagadze (2002). "Multi-dimensional vector product". arXiv:math.RA/0204357. {{cite journal}}: Cite journal requires |journal= (help)

[Jacobson-11] Nathan Jacobson (2009). Basic algebra I (Reprint of Freeman 1974 2nd ed.). Dover Publications. pp. 417–427. ISBN 0486471896.

[T_Smith-12] Further discussion of the tables and the connection of the Fano plane to these tables is found here: Tony Smith. "Octonion products and lattices". Retrieved 2010-07-11.

[Fauser-13] Rafał Abłamowicz, Bertfried Fauser (2000). Clifford Algebras and Their Applications in Mathematical Physics: Algebra and physics. Springer. p. 26. ISBN 0817641823.

[Manogue-14] Jörg Schray, Corinne A. Manogue (1996). "Octonionic representations of Clifford algebras and triality". Foundations of physics. 26 (1/January). Springer: 17–70. doi:10.1007/BF02058887. {{cite journal}}: Invalid |ref=harv (help) Available as ArXive preprint Figure 1 is located here.

[Abłamowicz0-15] Bertfried Fauser (2004). "§18.4.2 Contractions". In Pertti Lounesto, Rafał Abłamowicz (ed.). Clifford algebras: applications to mathematics, physics, and engineering. Birkhäuser. pp. 292 ff. ISBN 0817635254.

[Baez-16] John C. Baez (2001). "The Octonions" (PDF). Bull. Amer. Math. 39: 38. {{cite journal}}: Invalid |ref=harv (help)

[17] Elduque, Alberto (2004). "Vector cross products" (PDF). {{cite journal}}: Cite journal requires |journal= (help)

[18] Darpö, Erik (2009). "Vector product algebras". Bulletin of the London Mathematical Society. Vol. 41 (5): 898–902. doi:10.1112/blms/bdp066. {{cite journal}}: |volume= has extra text (help) See also Real vector product algebras

[k_vectors-19] Lounesto, §7.5: Cross products of k vectors in ℝⁿ, p. 98

[Gallier-20] Jean H. Gallier (2001). "Problem 7.10 (2)". Geometric methods and applications: for computer science and engineering. Springer. p. 244. ISBN 0387950443.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]