I think that the usual action of SO(3) on R^3 (defined by matrix multiplication) is faithful, because to non-identity rotations belong non-identity transformations.If we don't have originally a norm on R^3, but do have a faithful action of SO(3) on it, then we can try to define a norm by taking the length of an arbitrarily selected nonzero vector as 1, and the length of all vectors in its orbit space also as 1 (so, we declared the sphere of radius 1 in R^3). We define the length of the multiples of these unit vectors as the absolute value of the multiplier. The question is that is this norm well-defined? For example, is it sure that the action of a 180 degree rotation will bring all vectors to its negative? Because if not, then this definition is evidently ambiguous. Or, what extra conditions must this action satisfy to make this norm-definition unambiguous?