hcp packing

I am trying to play with fcc, bcc, hcp and radom style packing. All are working fine except hcp. For all other packing the unit cell is cubic and the unit vectors in all cases possess same direction. However, it is different in case of hcp.

As mentioned in manual, the unit cell vectors for hcp are 1 0 0, 0 sqrt(3) 0, 0 0 sqrt(8/3). This appears peculiar to me as it says that the vectors are all perpendicular to each other which should not be that case of hcp. However, over internet I have seen something like sqrt(2) sqrt(2/3) 0, sqrt(2) -sqrt(2/3) 0 and 0 0 sqrt(8/3) as the cell vectors.

Please clear the facts. Actually, I want to create hcp lattice with atom radius 0.005 m. Can anyone tell me what should be the appropriate scale value in the command line lattice hcp scale. The values that I have tried either created the lattice with overlapping spheres or sphere too far to each other resulting in very loose packing.