Finding the points of a cube given two points.

[ilikeintegrals]

Question:
A cube ABCD, has been placed somewhere in space and is cut by the xy plane. The z-axis indicates the height of the cube. We know that A = <10,7,4> and C = <9,5,6>, find B, D, A', B', C', D' (Where A', B', C', D' are the points which intersect with the xy plane. B and D have the same Z component. The origin is arbitrary.

Solution:
View from above.

OA = <10,7,4>
OB = <u, v, w>
OC = <9,5,6>
OD = <x, y, w>
AB = OB - OA = <u-10, v-7, w-4>
DC = OC - OD = <9 - x, 5 - y, 6 - w>
AB = DC
=> u-10 = 9-x
=> v-7 = 5-y
=> w-4=6-w, => w = 5

Using the equality of the other lines I get the same equations. I tried to construct a system of equations from a series of dot products (AB.BC = 0,BC.CD = 0, CD.DA = 0, DA.AB = 0), but mathematica is acting up and I don't think it can be solved. I also know that the length of the sides are all equal. But again, this gives a non-linear system of equations that can't be solved. I think that's all the information I can deduce. I am really stuck and would appreciate help.

 

[Spinnor]

This may be useless, but given the information you have sketch the top of the cube in a rectangular coordinate system with some care, square ABCD. Once I did the problem made sense and possible paths forward became apparent. For example a look at my sketch and I know z coordinates of B and D are the average of z coordinates A and C?