Mathematically, it's fine to say that you've lost the magnitude dimension. Before normalization, the vector lies in R^n, which is an n-dimensional manifold. A

Mathematically, it's fine to say that you've lost the magnitude dimension. Before normalization, the vector lies in R^n, which is an n-dimensional manifold. After normalization, the vector lies in the unit sphere in R^n, which is an (n-1)-dimensional manifold.