4 ms·
> Traditional vector math gives the output of the cross-product of two vectors the same type. (That's why it's only meaningful in 3d.) That is not really the r
by fiforpg 3y ago
> Traditional vector math gives the output of the cross-product of two vectors the same type. (That's why it's only meaningful in 3d.)
That is not really the reason. Relevant trivia: that cross product is defined only in dimensions 3 and 7 (not only in 3) has to do with existence of certain algebraic objects — division algebras — which only exist in dimensions 1 (reals), 2 (complex), 4 (quaternions), and 8 (octonions).
https://en.wikipedia.org/wiki/Seven-dimensional_cross_product https://en.wikipedia.org/wiki/Seven-dimensional_cross_produc...
https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras) https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(compositi...
You could say there is cross product in 1d by analogy, but of course it is always equal to 0.
- ComplexSystems 3y agoThe wedge product (or "outer product" in geometric algebra) generalizes the cross product and works in any number of dimensions, in the sense that the cross product in 3D is just the Hodge star of the wedge product. The cross product gives a vector perpendicular to the plane generated by two vectors, whereas the wedge product gives an object representing that plane directly, and the Hodge star then mapping it to its orthogonal complement.
- itishappy 3y agoThanks, my math's never been particularly rigorous. The 7 dimensional cross product is fascinating, my brain really wants it to be 5 dimensional. I often use the cross product in 2d too, don't tell anybody.