It is easy to compute on vectors. It is possible, but complicated and unintuitive (I have written code to do it in the past), to compute it on multivectors.
I'm arguing in favor of most of GA's language. I just keep finding that the wedge and inner product parts are fantastic, and the geometric product part isn't. And I think the reason people keep finding GA appealing is because they didn't have the wedge product before, so having that in their conceptual toolkit fixes a lot, while having the geometric product doesn't fix much on top of it. Anyway I've studied GA a lot and I still have basically no idea what 'AB' means when both are arbitrary-grade multivectors, and as far as I can tell most sources don't even try to explain it.
I'm arguing in favor of most of GA's language. I just keep finding that the wedge and inner product parts are fantastic, and the geometric product part isn't. And I think the reason people keep finding GA appealing is because they didn't have the wedge product before, so having that in their conceptual toolkit fixes a lot, while having the geometric product doesn't fix much on top of it. Anyway I've studied GA a lot and I still have basically no idea what 'AB' means when both are arbitrary-grade multivectors, and as far as I can tell most sources don't even try to explain it.