Is there some geometry in which a coordinate transformation of a vector of magnitude zero transforms to a vector that does not have a zero magnitude? Since the formula for the magnitude of a vector is √(x12+x22+...xn2), I can see no way for it to have magnitude zero unless every component is zero. Therefore it has to be the zero vector. Furthermore, since vectors are independent of the coordinate system they are in in Euclidean geometry, even if a coordinate transformations change coordinates, it seems to me a zero vector must contain the same coordinates of 0 all the way through. But if that is true, is there some weird geometry where it doesn't hold? In which a vector of zero magnitude transforms to a vector whose magnitude is not zero?