What is the Vector Equation for Finding the Vertices of a Parallelepiped?

[rainyheleny]

Hi there, not homework as such but a problem I've been scraping my brain over all day, sure there's a straightforward answer I'm missing! If anyone can put me out of my misery that would be great.

Homework Statement

A parallelepiped is constructed of 6 planes (obviously 3 parallel pairs) - for each of these planes I have a vector describing the normal to the plane, and a co-ordinate at which the normal intersects the plane. I need to find the co-ordinates of all 8 vertices of the parallelepiped.

2. The attempt at a solution

The solution I can see is to construct the equation for each plane and then solve for the points at which the planes intersect in sets of 3. This would involve 8 sets of 3-variable simultaneous equations. I'm able to do this in principle, but I suspect there's an easy rule or symmetry that I'm missing (or have forgotten) which will make this less laborious. Any ideas?! My apologies for the general nature of the problem, but I've searched around and don't seem to be able to find what I'm looking for. Perhaps I'm wrong and the 8 sets of equations is the only way through this!

Many thanks for any answers either way!

 

[haruspex]

Not sure this is any easier, but if v is a vertex, let the normals to the planes containing it be n₁ .. n₃ and the corresponding locator vectors (points in the planes) be l₁ ... l₃. Can you write a vector equation using v, n₁ and l₁?