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

  • Thread starter rainyheleny
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In summary, the problem is to find the coordinates of all 8 vertices of a parallelepiped. The suggested solution involves constructing equations for each plane and solving for the points of intersection, but the asker is wondering if there is an easier way. They have tried searching for a solution but have not found one. They also propose using a vector equation to find the coordinates.
  • #1
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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  • #2
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?
 

1. What is a parallelepiped?

A parallelepiped is a three-dimensional shape that has six parallelogram faces. It is similar to a box, but with slanted edges instead of right angles.

2. How many vertices does a parallelepiped have?

A parallelepiped has eight vertices, which are the points where three edges meet.

3. What is the relationship between the number of edges and vertices in a parallelepiped?

A parallelepiped has 12 edges, which is twice the number of vertices. This is because each edge is shared by two vertices.

4. How can you determine the coordinates of the vertices of a parallelepiped?

The coordinates of the vertices of a parallelepiped can be determined by finding the intersection points of the three pairs of parallel planes that make up the faces of the parallelepiped.

5. Can a parallelepiped have all equal sides and angles?

Yes, a parallelepiped can have all equal sides and angles. This type of parallelepiped is known as a cube.

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