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.
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No. Imagine strectching the plane in the $x$-direction. Formally $(x,y)\mapsto (2x,y)$. The angle $(1,1)$ makes with $(1,0)$, which, originally is 45 degrees, is reduced.
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).