Given 8 cube vertices and a point inside, find two points that a line going from inside point with vector V cuts the surface of the cube. I have the coordinates of a vertices of a cube and a point inside the cube. Now I draw a line from that point with vector V. How can I  find the coordinates in which that line cuts the cube surface. 

Let $p$ be the point inside the cube, and $\textbf{v}$ the vector.
For each of the six faces of the cube, find the point where the ray from $p$
in direction $\textbf{v}$ intersects the plane. The ray can be represented as $p + t \,\textbf{v}$, where $t \ge 0$ is a parameter. Solve for the $t$ that places
$p + t \,\textbf{v}$ on each plane. Discard the negative $t$ solutions.
The smallest positive $t$ solution represents the first plane hit by the ray.
This is the point you seek.
