Chapter 13.3, Problem 77E

Hexagonal sphere packing Imagine three unit spheres (radius equal to 1) with centers at O(0, 0, 0), $P(\sqrt{3},-1,0)$, and $Q(\sqrt{3},1,0)$. Now place another unit sphere symmetrically on top of these spheres with its center at R (see figure).

1. a. Find the coordinates of R. (Hint: The distance between the centers of any two spheres is 2.)
2. b. Let r_ij be the vector from the center of sphere I to the center of sphere J. Find r_OP, r_OQ, r_PQ, r_OR, and r_PR.