You may have heard someone say that cross-products exist only in 3-dim and 7-dim. But of course, this depends on what equations you want your cross-product to satisfy. For the 3-dim and 7-dim cases one usually wants a bilinear product of two vectors, which produces an orthogonal vector and satisfies \(|a\times b|^2=|a|^2|b|^2-(a\cdot b)^2\). The solutions could be to take the imaginary parts (i.e. non-scalar) of quaternions and octonions, and \(a\times b=\frac12(ab-ba)\) and \(a\cdot b=-\frac12(ab+ba)\) and \(|a|^2=-a^2\) , which explains the 3-dim and 7-dim. But what if you consider higher dimensional Cayley-Dickson numbers (sedenions and above)? It turns out they work, with the small modification \(|a\times b|^2=|a b|^2-(a\cdot b)^2\). For example the result of two 15-dim vectors would still be orthogonal to the inputs and the mentioned generalization always holds for dimensions \(2^n-1\). But only for quaternions and octonions you get that \(|ab|^2=|a|^2|b|^2\). A small caveat for the general case is, that due to zero divisors the higher dimensional product can be zero even for non-parallel vectors.