Word Count : 534

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To show that an arbitrary reciprocal lattice vector \(\vec{G} = h \vec{a}_1 + k \vec{a}_2 + l \vec{a}_3\) is perpendicular to the family of planes denoted by \((h,k,l)\) in the direct lattice space, we need to demonstrate that the scalar product between \(\vec{G}\) and any vector parallel to the planes of the family \((h, k, l)\) is zero. ### Step 1: Reciprocal Lattice Vectors In a crystal lattice, the reciprocal lattice vectors \(\vec{a}_1\), \(\vec{a}_2\), and \(\vec{a}_3\) are defined such that: \[ \vec{a}_1 \cdot \vec{a}_2 = \vec{a}_1 \cdot \vec{a}_3 = \vec{a}_2 \cdot \vec{a}_3 = 0 \] and \[ \vec{a}_1 \cdot \vec{a}_1 = a_1^2, \quad \vec{a}_2 \cdot \vec{a}_2 ___ _________ ____ _________ ________ _________ ______.
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