My stance is: the wedge product is the fantastic mathematical tool you are looking for, to make sense of rotations and generalizations of complex numbers to hig

My stance is: the wedge product is the fantastic mathematical tool you are looking for, to make sense of rotations and generalizations of complex numbers to higher dimensions, and the "geometric product" (of Hestenes) is not that useful, and almost every time it is useful it's useful because it's got the wedge product as a special case. I think that the vector calculus that should be used in physics should consist of "wedge products and interior products" as the fundamental operations, and the 'geometric product' should probably not be mentioned. That said, this is only my current judgment of the situation, which may change someday. I'm waiting to be convinced that the geometric product is so useful. I just find it so very cludgy when, for instance, GA texts are filled with use of 'grade projection" : take a geometric product with all its terms and then project off the grade you're interested in. It seems like the wrong tool is being used, and almost every time there's a multivector of 'mixed grade', it's regrettable. (Indeed, wedge products are used everywhere already, in hiding -- integration, determinants, matrix minors, commutators, cross products, rotation matrices, projective geometry, grassmann algebra, ...)