Exercise 54

If \(\mathbf{v}_1, \mathbf{v}_2,\) and \(\mathbf{v}_3\) are noncoplanar vectors, let \[ \mathbf{k}_1 = \frac{\mathbf{v}_2 \times \mathbf{v}_3}{\mathbf{v}_1 \cdot (\mathbf{v}_2 \times \mathbf{v}_3)} \quad \mathbf{k}_2 = \frac{\mathbf{v}_3 \times \mathbf{v}_1}{\mathbf{v}_1 \cdot (\mathbf{v}_2 \times \mathbf{v}_3)} \quad \mathbf{k}_3 = \frac{\mathbf{v}_1 \times \mathbf{v}_2}{\mathbf{v}_1 \cdot (\mathbf{v}_2 \times \mathbf{v}_3)} \] (These vectors occur in the study of crystallography. Vectors of the form \(n_1\mathbf{v}_1 + n_2\mathbf{v}_2 + n_3\mathbf{v}_3\), where each \(n_i\) is an integer, form a lattice for a crystal. Vectors written similarly in terms of \(\mathbf{k}_1, \mathbf{k}_2,\) and \(\mathbf{k}_3\) form the reciprocal lattice.) (a) Show that \(\mathbf{k}_i\) is perpendicular to \(\mathbf{v}_j\) if \(i \ne j\). (b) Show that \(\mathbf{k}_i \cdot \mathbf{v}_i = 1\) for \(i = 1, 2, 3\). (c) Show that \(\mathbf{k}_1 \cdot (\mathbf{k}_2 \times \mathbf{k}_3) = \frac{1}{\mathbf{v}_1 \cdot (\mathbf{v}_2 \times \mathbf{v}_3)}\).