Does linear transformation,
prevent preserve angle between two vectors? I guess that it is true, so if we translate normal vector of a plan, it will be orthogonal to translated plan.
Only a subset of linear transformations also preserves angles. Orthogonal transformations preserve length and angles and can easily be characterized. If you want to drop the length condition then also stretching with the same factor along all coordinate axes is allowed. Note that there are also non-linear angle preserving transformations (conformal maps).