Chapter 12.4, Problem 54E

If $v_1$, $v_2$, and $v_3$ are noncoplanar vectors, let

$k_1=\frac{v_2\times v_3}{v_1\cdot \left(v_2\times v_3\right)}$ $k_2=\frac{v_3\times v_1}{v_1\cdot \left(v_2\times v_3\right)}$

$k_3=\frac{v_1\times v_2}{v_1\cdot \left(v_2\times v_3\right)}$

(These vectors occur in the study of crystallography. Vectors of the from $n_1{v}_1+n_2{v}_2+n_3{v}_3$, where each $n_i$ is an integer, from a lattice for a crystal. Vectors written similarly in terms of $k_1,k_2$, and $k_3$ from the reciprocal lattice.)

(a) Show that $k_i$ is perpendicular to $v_j$ if $i\ne j$.

(b) Show that $k_i\cdot v_i=1$ for $i=1,2,3$.

(c) Show that $k_1\cdot \left(k_2\times k_3\right)=\frac{1}{v_1\cdot \left(v_2\times v_3\right)}$.