Can you define something like cross product of three-dimensional vectors in other dimensions? That depends on what you mean by a product being “like” the familiar cross product. In [1] Walsh lists several properties of the cross product, and proves that there exists a product with these properties only in dimensions 1, 3, and 7.
Walsh points out that given the first four properties, the last is equivalent to saying that the product of two orthogonal unit vectors is another unit vector.
In one dimension, define the cross product of any two “vectors” to be 0. This trivial cross product only works in one dimension because there are no orthogonal unit vectors in one dimension.
To define his cross product in seven dimensions, Walsh partitions the seven components of a vector into groups of 3, 1, and 3 and treats the groups of size three as three dimensional vectors. Then he defines the product as follows:
On the left side, × means the cross product in seven dimensions. On the right side, × means the ordinary cross product in three dimensions.
Although this product satisfies the properties of cross product given at the top of the post, Walsh points out that his product does not satisfy the Lie identity
that the ordinary cross product satisfies. This shows that the Lie identity is not a consequence of the identities above.
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[1] Bertram Walsh. The Scarcity of Cross Products on Euclidean Spaces. The American Mathematical Monthly, Feb., 1967, pp. 188-194