5) Use the 3-index system to 4-index system formula to convert it and eventually get [11 (-2) 1] for the mentioned vector. However, this would imply that the z vector is independent of the ratio in a1 and a2 since using this method we would have gotten 0.5 unit length for a1 and a2 and 1 unit length for z. If not, shouldn’t it be [11 (-2) 2] instead? I’m sorry if I sound confusing, because I’m very confused myself as well. But if anyone knows about this, please help, thanks! I am not sure what one means by the direction vector. There is a unique planar orientation, the basal pole (normal to the basal plane), which is in the c-direction of the hexagonal (hcp) crystal. Then there are the three a-directions. Besides that, there are prismatic and pyramidal planes and associated directions. Source.