Line plane intersection

I have two planes in $\mathbb R^3$ as shown below:

axes representation corrected after MvG's comment

Each plane is a finite area, a rectangle with length and width $H_l, H_w$.Each plane has its own coordinate system, with $x,y$ axes parallel to plane edges and origins of $C1_, C_2$ at $ plane_1 , plane_2$ respectively.

Two points $P,e$ belong to plane 1 and plane 2 respectively.Each point is described with respect to the plane's coordinate system.

Point $e(x_e, y_e)$ on $plane_2$ represents the intersection of the line which passes from point $P(x_p, y_p)$ of $plane_1$ and has the direction vector $\overrightarrow S = <S_x, S_y, S_z>$.

Which are the "e" point coordinates ($x_e$,$y_e$) with respect to the coordinates system of the plane it belongs ($plane_2$)?

I have the following data available:

1. Centers of planes: $$C_1(x_1,y_1,z_1), C_1(x_2,y_2,z_2)$$ with respect to the global coordinate system
2. Plane normals: $$\hat n_1 = <n_{x_1}, n_{y_1}, n_{z_1}>, \hat n_2 = <n_{x_2}, n_{y_2}, n_{z_2}>$$
3. Direction vector: $$\overrightarrow S = <S_x, S_y, S_z>$$
4. Point $P(x_p,y_p)$ with respect to the coordinate system of $plane_1$
5. Also the azimuthial and elevation angles for both planes with respect to the global coordinate system $C(0,0,0)$ are known: $alpha_{H_1}, \alpha_{H_1}, alpha_{H_2}, \alpha_{H_2}$

In order to solve the problem:

1. I check if $\overrightarrow S \bullet \hat n_2 = 0$ in this case there is no intersection and point $e$ does not exist.-
2. While I know the $plane_2$ normal vector $\hat n_1$ and point $C_2$ I define the equation of $plane_2$ as: $n_x(x - x_2) + n_y(y - y_2) + n_z(z - z_2) = 0$
3. I write the parametric form of a line $\mathscr P$ with direction vector $\hat S$ that passes from $P(X_p, Y_p, Z_p)$, where $P(X_p, Y_p, Z_p)$ is point $P(x_p, y_p)$ transformed from the coordinate system of $plane_1$ to the global coordinate system (by the way how can I do this transformation?) :$$\mathscr P\begin{cases}x = X_p + S_x t \\y = Y_p + S_y t \\z = Z_p + S_z t\end{cases}$$
4. Then from the above equations $t$ can be found and subsequently point $e(X_e, Y_e, Z_e)$ with respect to the global coordinate system.
5. At last I have to transform point $e(X_e, Y_e, Z_e)$ from the global coordinate system to the coordinate system of $plane_2$

Is the above solution right?How can I transform the points between the coordinate systems>Is there any smarter/quicker way to find point $e(x_e,y_e)$?