Consider a tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 2, 0) and (0, 0, 3). (a) Compute the area of each face of the tetrahedron.
(b) Compute the normal vector to each of the four faces, that points outwards and has magnitude equal to the area of the corresponding face.
(c) Check that the sum of the four normal vectors you found is 0.
(d) This fact is true for any polyhedron: the sum of their normal vectors (with magnitude equal to the area of the corresponding face) is zero. The converse statement is also true: Given a finite set of vectors whose sum is zero, there is a polyhedron whose faces are normal to the given vectors and have areas equal to their magnitudes. Find online the name of the mathematician who proved this theorem and write 3 facts about his life.
(e) Use CalcPlot3D to plot the tetrahedron: i. Start with a blank graph. In "Add to graph" select Point. Enter the coordinates of a point and choose the color you want.