3D Vector Geometry

My first post, so would be very grateful if anyone could help...

I have a problem concerning the calculation of intercept points of lines on the surface of a cube.

Firstly consider a set of parallel lines that are heading towards a cube and travelling in the direction parallel to the x axis. The y and z separations of the intercept points on the y-z plane (x=0) of the cube are given by s.

I now need to consider the same set of lines, but travelling in an arbitrary direction and intercepting on any face of the cube. For a given intercept point, I need to find the (dx,dy,dz) distance, on the surface of the cube, to the intercepts of the adjacent lines.

Does anyone have any suggestions?