An architect is building a structure that will place vertical pillars at the vertices of a regular hexagon ABCDEF, which is lying horizontally on the ground. The 6 pillars hold up a flat solar panel that will not be parallel to the ground.the heights of the pillars at A, B and C are 12, 9 and 10 metres respectively. What is the height of the pillar at E.
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if we want a numerical answer, we need to assume ABCDEF is regular.* Let’s place it parallel to the x-y plane at z = -9. We’ll put B at the origin and E on the positive y-axis. Call the hexagon formed by the tops of the pillars A’B’C’D’E’F’. Then B’ is at the origin. It turns out the answer is independent of the length of the sides of ABCDEF but that wasn’t obvious to me so I let the length of each side be 2a. Then we have A’(a, sqrt(3)a, 3), C’(a, sqrt(3)a, 1) and E’(4a, 0, h).
A normal vector N to the plane of the solar panel will be given by the cross product of vectors B’A’ and B’C’: N = <4sqrt(3)a, 2a, -2sqrt(3)a^2>. The equation of the plane can be found by crossing N with vector B’P, where P is at (x, y, z). The equation is 4sqrt(3)ax + 2ay - 2sqrt(3)a^2z = 0. Substituting in point E’ we find h = 8 so the height of the pillar at E is 17. (And if anyone cares, the pilars at D and F have heights 14 and 16 respectively.)