Question

1. If \(\vec{v_1}\), \(\vec{v_2}\), and \(\vec{v_3}\) are noncoplanar vectors, let \(\vec{k_1} = \frac{\vec{v_2} \times \vec{v_3}}{\vec{v_1} \cdot (\vec{v_2} \times \vec{v_3})}\), \(\vec{k_2} = \frac{\vec{v_3} \times \vec{v_1}}{\vec{v_1} \cdot (\vec{v_2} \times \vec{v_3})}\), \(\vec{k_3} = \frac{\vec{v_1} \times \vec{v_2}}{\vec{v_1} \cdot (\vec{v_2} \times \vec{v_3})}\ (These vectors occur in the study of crystallography. Vectors of the form \(n_1\vec{v_1} + n_2\vec{v_2} + n_3\vec{v_3}\), where each \(n_i\) is an integer, form a lattice for a crystal. Vectors written similarly in terms of \(\vec{k_1}\), \(\vec{k_2}\), \(\vec{k_3}\). form the reciprocal lattice.) (a) Show that \(\vec{k_i}\) is perpendicular to \(\vec{v_j}\) if \(i \neq j\). (b) Show that \(\vec{k_i} \cdot \vec{v_i} = 1\) for \(i = 1, 2, 3\). (c) Show that \(\vec{k_1} \cdot (\vec{k_2} \times \vec{k_3}) = \frac{1}{\vec{v_1} \cdot (\vec{v_2} \times \vec{v_3})}\)

          1. If \(\vec{v_1}\), \(\vec{v_2}\), and \(\vec{v_3}\) are noncoplanar vectors, let
\(\vec{k_1} = \frac{\vec{v_2} \times \vec{v_3}}{\vec{v_1} \cdot (\vec{v_2} \times \vec{v_3})}\), \(\vec{k_2} = \frac{\vec{v_3} \times \vec{v_1}}{\vec{v_1} \cdot (\vec{v_2} \times \vec{v_3})}\), \(\vec{k_3} = \frac{\vec{v_1} \times \vec{v_2}}{\vec{v_1} \cdot (\vec{v_2} \times \vec{v_3})}\
(These vectors occur in the study of crystallography. Vectors of the form
\(n_1\vec{v_1} + n_2\vec{v_2} + n_3\vec{v_3}\), where each \(n_i\) is an integer, form a lattice for a crystal.
Vectors written similarly in terms of \(\vec{k_1}\), \(\vec{k_2}\), \(\vec{k_3}\). form the reciprocal lattice.)
(a) Show that \(\vec{k_i}\) is perpendicular to \(\vec{v_j}\) if \(i \neq j\).
(b) Show that \(\vec{k_i} \cdot \vec{v_i} = 1\) for \(i = 1, 2, 3\).
(c) Show that \(\vec{k_1} \cdot (\vec{k_2} \times \vec{k_3}) = \frac{1}{\vec{v_1} \cdot (\vec{v_2} \times \vec{v_3})}\)

1. If v⃗1⃗, v⃗2⃗, and v⃗3⃗ are noncoplanar vectors, let
k⃗1⃗ = (v⃗2⃗×v⃗3⃗)/(v⃗1⃗· (v⃗2⃗×v⃗3⃗)), k⃗2⃗ = (v⃗3⃗×v⃗1⃗)/(v⃗1⃗· (v⃗2⃗×v⃗3⃗)), k⃗3⃗ = (v⃗1⃗×v⃗2⃗)/(v⃗1⃗· (v⃗2⃗×v⃗3⃗))(These vectors occur in the study of crystallography. Vectors of the form
n1v⃗1⃗ + n2v⃗2⃗ + n3v⃗3⃗, where each ni is an integer, form a lattice for a crystal.
Vectors written similarly in terms of k⃗1⃗, k⃗2⃗, k⃗3⃗. form the reciprocal lattice.)
(a) Show that k⃗i⃗ is perpendicular to v⃗j⃗ if i ≠ j.
(b) Show that k⃗i⃗·v⃗i⃗ = 1 for i = 1, 2, 3.
(c) Show that k⃗1⃗· (k⃗2⃗×k⃗3⃗) = (1)/(v⃗1⃗· (v⃗2⃗×v⃗3⃗))

Added by Miguel -Ngel H.

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