I think there are two reasons why the product doesn't commute:
qp and pq are linear operations, and we know that rotation + translation composition is nonlinear.
We also know that 3D rotations aren't commutative as well.
So $$ pq = qp $$ iff translation is 0, and rotation is identity.
keenan
@lwan Good insight. Yes, this is a nice way to argue. Since you know quaternion multiplication can be used to encode 3D rotations, and 3D rotations don't commute, there's no way multiplication could commute (in general). Much easier then writing out the full product both ways. :-)
I think there are two reasons why the product doesn't commute:
So $$ pq = qp $$ iff translation is 0, and rotation is identity.
@lwan Good insight. Yes, this is a nice way to argue. Since you know quaternion multiplication can be used to encode 3D rotations, and 3D rotations don't commute, there's no way multiplication could commute (in general). Much easier then writing out the full product both ways. :-)