Q:

Tangent planes of a surface parallel to a fixed direction

I'm learning differential geometry, specifically the theory of surfaces, and need help with the following problem:

Consider the surface $S$ given by $x_1 = x_2 - f(x_2 - x_3)$, where $f \in C^2$.
Show that every tangent plane of $S$ are parallel to a fixed direction.

Here's my work so far:
We first need to find the parametric representation of $S$. Setting $x_2 = u$ and $x_3 = v$ give $x_1 = u - f(u - v)$. Therefore, the parametric representation of $S$ is given by
$$S : \vec x(u, v) = (u - f(u - v)) \vec e_1 + u \vec e_2 + v \vec e_3$$
or in a more conviennent/compact notation
$$S : \vec x(u, v) = (u - f, u ,v).$$
The equation of the tangent plane at $\vec x(u_P, v_P)$ in parametric form with parameters $\mu, \lambda$ is given by
$$\vec T_P(S) : \vec y = \vec x(u_P, v_P) + \lambda \vec x_u(u_P, v_P) + \mu \vec x_v(u_P, v_P) \tag{*}$$
where
$$\vec x_u(u_P, v_P) = (1 - f', 1, 0)$$
and
$$\vec x_v(u_P, v_P) = (f', 0, 1).$$
Substituting in $(*)$ yields
$$\vec T_P(S) : \vec y = (u - f, u ,v) + \lambda (1 - f', 1, 0) + \mu (f', 0, 1). \tag{**}$$

From $(**)$ I don't know how to prove that the tangent planes are parallel to a fixed direction. I'm also confused by the meaning of the phrase "parallel to a fixed direction". Does this means that the tangent planes of the surface are all parallel to some plane of $\mathbb R^3$?

A:

Given
$$
x_{\,1}  = x_{\,2}  + f(x_{\,2}  - x_{\,3} )\quad i.e.\quad F(x_{\,1} ,x_{\,2} ,x_{\,3} ) = 0
$$
we can rewrite it ,as
$$
x_{\,1}  - x_{\,3}  = x_{\,2}  - x_{\,3}  + f(x_{\,2}  - x_{\,3} )\quad  \Rightarrow \quad y_{\,1}  = y_{\,2}  + f(y_{\,2} )\quad i.e.\quad G(y_{\,1} ,y_{\,2} ) = 0
$$
or, otherwise, as
$$
x_{\,1}  - x_{\,2}  = f(x_{\,2}  - x_{\,3} )\quad  \Rightarrow \quad z_{\,1}  = f(z_{\,2} )\quad i.e.\quad H(z_{\,1} ,z_{\,2} ) = 0
$$
Now, the tranformations
$$
\left( {x_{\,1} ,x_{\,2} ,x_{\,3} } \right) \Leftrightarrow \left( {y_{\,1} ,y_{\,2} ,y_{\,3} } \right)\quad \left( {x_{\,1} ,x_{\,2} ,x_{\,3} } \right) \Leftrightarrow \left( {z_{\,1} ,z_{\,2} ,z_{\,3} } \right)
$$
with the understanding that $y_3$ and $z_3$ are vectors independent from other two, so as to complete the respective basis,
are linear transformations, by which planes transform into planes, and "tangent to $F$" into "tangent to $G$" / "tangent to $H$".
$G$ and $H$ are clearly cylinders, with axes $y_3$ and $z_3$ respectively, so their tangent planes will all be parallel to those directions.
In the Euclidean space such a direction would be unique, that of $(1,1,1)$.

