Example
The postulates underlying construction of the seven-dimensional cross product are presented in the section Definition. As context for that discussion, the historically first example of the cross product is tabulated below using e1 to e7 as basis vectors. This table is one of 480 independent multiplication tables fitting the pattern that each unit vector appears once in each column and once in each row. Thus, each unit vector appears as a product in the table six times, three times with a positive sign and three with a negative sign because of antisymmetry about the diagonal of zero entries. For example, e1 = e2 × e3 = e4 × e5 = e7 × e6 and the negative entries are the reversed cross products.
| Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Letter | i | j | k | l | il | jl | kl |
| Alternate | i | j | k | l | m | n | o |
| × | e1 | e2 | e3 | e4 | e5 | e6 | e7 |
|---|---|---|---|---|---|---|---|
| e1 | 0 | e3 | −e2 | e5 | −e4 | −e7 | e6 |
| e2 | −e3 | 0 | e1 | e6 | e7 | −e4 | −e5 |
| e3 | e2 | −e1 | 0 | e7 | −e6 | e5 | −e4 |
| e4 | −e5 | −e6 | −e7 | 0 | e1 | e2 | e3 |
| e5 | e4 | −e7 | e6 | −e1 | 0 | −e3 | e2 |
| e6 | e7 | e4 | −e5 | −e2 | e3 | 0 | −e1 |
| e7 | −e6 | e5 | e4 | −e3 | −e2 | e1 | 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
where is a completely antisymmetric tensor with a positive value +1 when ijk = 123, 145, 176, 246, 257, 347, 365. By picking out the factors leading to the unit vector e1, for example, one finds the formula for the e1 component of x × y. Namely
The top left 3 × 3 corner of the table is the same as the cross product in three dimensions. It also may be noticed that orthogonality of the cross product to its constituents x and y is a requirement upon the entries in this table. However, because of the many possible multiplication tables, general results for the cross product are best developed using a basis-independent formulation, as introduced next.
Read more about this topic: Seven-dimensional Cross Product
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“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
—Marcel Proust (1871–1922)