Source
- URL: https://x.com/HostOfMeta/status/2093065798399176704
- Author: Jeremie Pelletier (@HostOfMeta)
- Posted: 2026-08-27 19:59:08
Thread
1/ @HostOfMeta
After 2 decades of using it naively, I have deconstructed the (×) cross product. Only took learning geometric algebra. Goes like this:
u×v = ⋆(u∧v) = w
As in, it combines two operations: a lift and a drop. First the wedge creates a bivector out of (u) and (v), which turns the two vectors into an oriented plane. Then the dual multiplies by the antiscalar, which cancels out (u) and (v) while leaving an orthogonal (w) vector in place.
And explains why it only works in 3D (7D uses a different contraction, because this formula yields a 5D blade.)
2/ @HostOfMeta
Quick precision: the “cancels out” part only applies when (u) and (v) are the basis vectors e1 and e2, meaning their coefficients stay unitary as (1).
When these input vectors have non-unit coefficients, then the dual makes more of a selection operation because they play a role.