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

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SUMMARY

The discussion centers on deriving the vector equation for finding the vertices of a parallelepiped, defined by six planes with known normal vectors and intersection coordinates. The solution involves constructing equations for each plane and solving for the intersection points of sets of three planes, resulting in eight sets of simultaneous equations. The participants suggest that there may be a simpler method or symmetry that can streamline this process. The discussion emphasizes the need for a clear understanding of vector equations and their applications in geometry.

PREREQUISITES
  • Understanding of vector equations and their components
  • Knowledge of simultaneous equations and their solutions
  • Familiarity with geometric concepts related to planes and intersections
  • Basic proficiency in linear algebra
NEXT STEPS
  • Research the properties of parallelepipeds and their geometric characteristics
  • Learn about vector equations and their applications in 3D geometry
  • Study methods for solving systems of simultaneous equations
  • Explore symmetry principles in geometric shapes to simplify calculations
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Mathematicians, geometry enthusiasts, engineering students, and anyone involved in computational geometry or vector analysis will benefit from this discussion.

rainyheleny
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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!
 
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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?
 

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