Word Count : 632

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To solve this problem, we need to determine two things: 1. The volume of the primitive cell. 2. The reciprocal lattice vectors. ### 1. Volume of the Primitive Cell The volume of the primitive cell in a lattice is given by the scalar triple product of the lattice vectors: \[ V = |\vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3)| \] where \( \vec{a}_1 \), \( \vec{a}_2 \), and \( \vec{a}_3 \) are the primitive lattice vectors. From the given vectors: \[ \vec{a}_1 = 2\hat{i} - \hat{j}, \quad \vec{a}_2 = 2\hat{i} + \hat{j}, \quad \vec{a}_3 = 6\hat{k} \] We calculate the cross product \( \vec{a}_2 \times \vec{a}_3 \): \[ \vec{a}_2 \times \vec{a}_3 = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & 0 \\ 0 & 0 _____ _______ ______ ______ _________ ________ _____.
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