I am trying to understand how to do the cross product of two four-dimensional vectors. From what I understood it's not possible, unless the vectors are of the form
$\mathbb R^3 \times \{0\}$ or $\{0\} \times \mathbb R^3$.
So for example, if I have $u = (1, \frac12, 0, 0)$, $v = (0, 2, 1, 0)$, what is the cross product $w = u \times v$?
Best Answer
Four-dimensional Euclidean space does not have a binary cross product. (If it did, you could use it to define a five-dimensional division algebra, which isn't possible.) However, if you choose a three-dimensional subspace of $\mathbb{R}^4$, then that subspace inherits the inner product, which uniquely determines a cross product (up to choice of orientation).
You seem to be talking about $\mathbb{R}^3\times\{0\}$ as a 3D subspace of $\mathbb{R}^4$, in which case to calculate the cross product of two vectors (in this 3D subspace) you simply ignore the fourth coordinate (which is $0$) and do the calculation with the first three coordinates.
There is a ternary cross product on $\mathbb{R}^4$ in which you can compute a vector perpendicular to three given ones, with size and orientation based on the parallelotope generated by the three vectors (instead of a parallelogram as with two vectors). This can be calculated with differential forms if one was so inclined.