?If \(v_{1}, v_{2}\), and \(v_{3}\) are noncoplanar vectors, let \(k_{1}=\frac{v_{2}

Chapter 12, Problem 54

(choose chapter or problem)

If \(v_{1}, v_{2}\), and \(v_{3}\) are noncoplanar vectors, let

        \(k_{1}=\frac{v_{2} \times v_{3}}{v_{1} \cdot\left(v_{2} \times v_{3}\right)} \quad k_{2}=\frac{v_{3} \times v_{1}}{v_{1} \cdot\left(v_{2} \times v_{3}\right)}\)

                   \(k_{3}=\frac{v_{1} \times v_{2}}{v_{1} \cdot\left(v_{2} \times v_{3}\right)}\)

(These vectors occur in the study of crystallography. Vectors of the form \(\mathbf{n}_{1} \mathbf{v}_{1}+\mathbf{n}_{2} \mathbf{v}_{2}+\mathbf{n}_{3} \mathbf{v}_{3}\) , where each ni is an integer, form a lattice for a crystal. Vectors written similarly in terms of \(\mathbf{k}_{1}, \mathbf{k}_{2}\), and \({k}_{3}\) form the reciprocal lattice.)

(a) Show that \(\mathbf{k}_{\mathrm{i}}\) is perpendicular to \(\mathbf{v}_{\mathrm{j}}\) if \(i \neq j\).

(b) Show that \(\mathbf{k}_{\mathrm{i}} \mathbf{v}_{\mathrm{i}}=1\) for \(\mathbf{i}=1,2,3\).

(c) Show that \(\mathbf{k}_{1}\left(\mathbf{k}_{2} \times \mathbf{k}_{3}\right)=\frac{1}{v_{1} \cdot\left(v_{2} \times v_{3}\right)}\).

Equation Transcription:

v1, v2 

v3

 

n1v1 + n2v2 + n3v3

k1, k2,

k3

vj

i  j

ki vi

i = 1, 2, 3

k1  (k2  k3 ) =

Text Transcription:

v_1, v_2

v_3

k_1=v_2 times v_3/v_1 times (v_2 times v_3)

k_2=v_3 times v_1/v_1 times (v_2 times v_3)

k_3=v_1 times v_2/v_1 times (v_2 times v_3)

n_1v_1 + n_2v_2 + n_3v_3

k_1, k_2,

k_3

v_j

i neq j

k_i v_i

i = 1, 2, 3

k_1  (k_2 times k_3 ) = 1/v_1 times (v_2 times v_3)