[extasia]

Magnitude is not a dimension, it’s information about each value that is lost when you normalize it. To prove this normalize any vector and then try to de-normalize it again.

[kccqzy]

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.)

[immibis]

Impossible because... you lost a dimension.

[extasia]

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?

[ekianjo]

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