6. The columns of a regular hexahedron shown in Figure 1 are listed as columns i

Question

6. The columns of a regular hexahedron shown in Figure 1 are listed as columns in the matrix below. Vertex A is in the first column, B in the second column, etc.

A B C D E F

1 -1 0 0 0 0

0 0 1 -1 0 0

0 0 0 0 1 -1

Find the distance between two opposite faces, such as AEC and BDF. Hint: one way to do this to translate the figure so that one of the faces passes through the origin and orthogonally project a point from the opposing face to the face passing through the origin.

Shape looks like this: http://images.yourdictionary.com/hexahedron

Explanation / Answer

B = (-1, 0, 0), D = (0,-1,0), F = (0,0,-1)

A = (1,0,0),C = (0,1,0),E=(0,0,1)

We know that opposite faces of such a regular hexahedron as shown are parallel.So as mentioned in the hint we shall translate so that one of the faces,say BDF,passes through the origin.

(X,Y,Z) = (x,y,z) + (1,0,0) Here small letters for old coordinates capitals for new coordinates.

Then B' = (0,0,0), D' = (1,-1,0), F'= (1,0,-1), A' = (2,0,0)

Now we shall take point A' which is on the opposite face and do orthogonal projection on to this plane.

Clearly the equation of the plane B'D'F' is : x+y+z = 0

so unit normal to the plane is n = (1,1,1)/root 3

hence orthogonal projection of A' on to the plane B'D'F'

=<n,B'A'> = < [(1,1,1) /root 3], (2,0,0)> = 2/root 3

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