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Suppose I have an arbitrary set of Miller indices $(hkl)$ for a 3-d crystal lattice defined by the primitive vectors $\{\vec{a}_i\}$ for $i=1,2,3$. We know that the plane labeled by $(hkl)$ must always intersect the crystal axes at points $\frac{\vec{a}_1}{h}$, $\frac{\vec{a}_2}{k}$, and $\frac{\vec{a}_3}{l}$. It is not always the case that $[hkl]$ is a vector normal to the plane.

However, when the lattice is a simple cubic lattice, it happens to be that $[hkl]$ is indeed perpendicular to the plane. I was trying to think about under what conditions the lattice type has to satisfy for this to hold, more generally; but I couldn't exactly wrap my head around it. Would anyone have some thoughts to share?

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