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).
a. Find the coordinates of $R$. (Hint: The distance between the centers of any two spheres is 2.)
b. Let $\mathbf{r}_{i j}$ be the vector from the center of sphere $i$ to the center of sphere $j .$ Find $\mathbf{r}_{O P}, \mathbf{r}_{O Q}, \mathbf{r}_{P Q}, \mathbf{r}_{O R},$ and $\mathbf{r}_{P R}$.