I found this problem in Christman's book fundamental of solid state physics

What is the Miller index for a plane parallel to both $3\vec{a}+\vec{c}$ and $\vec{b}$, in any lattice?


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When a plane is parallel to two vectors, the dot product between its normal and that of the vectors is zero (they are perpendicular). So you are looking for a vector that is perpendicular to both $3\vec a + \vec c$ and $\vec b$.

The answer, by inspection, is the vector $-\vec a + 0 \vec b + 3 \vec c$, from which the Miller indices are (if I remember this correctly) $(\bar 1 0 3)$


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