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Cross products likely won't vanish, precisely because they're heavily linked to the Clifford algebra, and are essentially all you need for the most common case.
by contravariant 10y ago
Cross products likely won't vanish, precisely because they're heavily linked to the Clifford algebra, and are essentially all you need for the most common case.
By the way, it is possible to generalize the cross product to higher dimensions, it just won't be a nice binary product, in n dimensions you need n-1 vectors to uniquely determine one orthogonal to all of them. Of course you can achieve pretty much the same thing with the Clifford algebra, or with wedge products.
- ittekimasu 10y agoThis is a question of what it "means". It's origins can be traced to problems dealing with SO(n), and in this sense it's uniquely defined by the Lie-algebra of \mathfrak{so}(n), but in the Clifford-algebra world they become blades (more appropriate for CG). I consider the cross-product to be one of the most abused objects in Physics; my first course in Newtonian mechanics was a nightmare, filled as it were, with the ghosts of all the characters killed so as to make the plot ostensibly "simple".