I've been interested in (U)GA and related subjects for a while. Looks really promising and its advocates make it seem like it solves all kinds of problems. But I hesitated to dive in because the learning curve looks rather steep, and it's not clear if the cost-benefit of ascending that curve ends one up in the black.
How useful is (U)GA, really? In other words, what are the concrete benefits relative to traditional vectors, matrices, quaternions? Are those benefits worth the additional complexity (or is GA actually a simplification)? Are there applications where GA is the clear win, or is it simply an alternative way of expressing what can already be expressed through the 3 primitives I mentioned?
On another tack, how would you characterize the power / utility of (U)GA? For instance, it _seems_ like (U)GA is a super-powerful tool that's just misunderstood and everyone would benefit from learning it (like Lisp in the early 2000s). On the other hand, papers often extol the virtues of (U)GA by showing how it can solve (seemingly) esoteric higher-order math problems.