$1\mathrm{.}1$ Four rectangular parallelepipeds, $A,B,C,$ and $D,$ are arranged as shown in

Fig. P$1\mathrm{.}1,$ where $a_1,a_2,a_3$ are unit vectors parallel to edges of $A,b_1,b_2,b_3$ are

unit vectors parallel to edges of $B,$ and so forth, while $q_1,q_2,$ and $q_3$ are the radian

measures of angles that determine the relative orientations of $A,$dots,$D.$ The

configuration shown is one for which $q_1,q_2,$ and $q_3$ are positive.

Figure P$1\mathrm{.}1$