Magnitude is a dimension. Any 2-dimensional vector can be explicitly transformed into the polar (r, theta) coordinate system where one of the dimensions is magnitude. Any 3-dimensional vector can be transformed into the spherical (r, theta, phi) coordinate where one of the dimensions is magnitude. This is high school mathematics. (Okay I concede that maybe the spherical coordinate system isn't exactly high school material, then just think about longitude, latitude, and distance from the center.)

Impossible because... you lost a dimension.

That’s not mathematically accurate though, is it? We haven’t reduced the dimension of the vector by one.

Pray tell, which dimension do we lose when we normalize, say a 2D vector?

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.

Magnitude, obviously.

>>> Magnitude is not a dimension [...] To prove this normalize any vector and then try to de-normalize it again.

Say you have the vector (18, -5) in a normal Euclidean x, y plane.

Now project that vector onto the y-axis.

Now try to un-project it again.

What do you think you just proved?

A circle circumference is a line, is 1D?

you dont lose anything when you normalize things. not sure what you are tallking about.