I'm just curious what the historical significance of using h,k in vector form are. It's very likely that the answer is, there is no significance just like there is no significance to using x,y,z.
Using h,k just seems kind of a reach for me like there must be a reason. Maybe from the original latin or something? Any clues?
I'm pretty sure it comes from the notation used for quaternions, which as a vector space the underlying set is defined as $$\mathbb{H}=\{a+bi+cj+dk\quad |\quad a,b,c,d\in \mathbb{R}\},$$
and $i,j,$ and $k$ represent the imaginary units all squaring to $-1$ (there are relations among the $i,j$ and $k$ but they are not necessary here). Thus the (purely) imaginary quaternions are isomorphic to $\mathbb{R}^3$, with linearly independent unit vectors labeled by $i,j$ and $k$ which point in the three independent imaginary directions of the quaternions.