(a) Given an arbitrary (i.e., not necessarily regular) tetrahedron, associate to each of...

(a) Given an arbitrary (i.e., not necessarily regular) tetrahedron, associate to each of its four triangular faces a vector outwardly normal to that face with length equal to the area of that face. (See Figure 1.117.) Show that the sum of these four vectors is zero. (Hint: Describe v₁, . . . , v₄ in terms of some of the vectors that run along the edges of the tetrahedron.)

(b) Recall that a polyhedron is a closed surface in R³ consisting of a finite number of planar faces. Suppose you are given the two tetrahedra shown in Figure 1.118 and that face ABC of one is congruent to face A’B’C’ of the other. If you glue the tetrahedra together along these congruent faces, then the outer faces give you a six-faced polyhedron. Associate to each face of this polyhedron an outward-pointing normal vector with length equal to the area of that face. Show that the sum of these six vectors is zero.

(c) Outline a proof of the following: Given an n-faced polyhedron, associate to each face an outward-pointing normal vector with length equal to the area of that face. Show that the sum of these n vectors is zero.