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Why is u x (v x w) the altitude of the triangle that hits the base represented by u? I'm having a tough time visualizing this.


I was trying to think of the geometric interpretation but could not. As far as I know, given LaGrange's identity, I think that the cross product of three vectors is the volume of their respective parallelepiped. However, I can't really visualize the right hand side of the identity. I'm thinking that it could be a different way to calculate the volume...


In what circumstances we can use these identities in CG?


I'm a little confused about the correspondence between the provided diagram and the Jacobi identity. Is it illustrating a special case of some kind? The identity seems to hold for arbitrary u, v, and w whose lengths, in general, need not satisfy the triangle inequality.


In v(u dot w), what is the operator between v and (u dot w)?


From a visualization perspective, is the Jacobi Identity true because each of the terms, e.g. u x (v x w), represents a vector stemming out from the triangle (basically reflect each altitude against the side to which it is perpendicular)?


How is the Jacobi identity working in this illustration? I can see that (v x w) would be orthogonal to the triangle, and then u x (v x w) would be perpendicular to u, but why is the result 0 when you sum all three?