here is a question on reciprocal lattices that im stuck on for a simple cubic lattice, the unit cell is defined by a1=a(1,0,0) a2 = a(0,1,0) a3 = a(0,0,1), demonstrate that the reciprocal lattice of its reciprocal lattice is the original crystal lattice. From what ive found, i think the reciprocal lattice base vectors b1 b2 b3 of the primitive vectors of the crystal lattice a1 a2 a3 is defined by (π is pi btw) b1=2π (a2xa3/a1.a2xa3), b2=2π (a3xa1/a1.a2xa3) , b3=2π(a1xa2/a1.a2xa3) the volume V is defined by a1.a2xa3 so i have to figure that out, but if i were to do a2xa3, would it be: i j k 0 1 0 0 0 1 = (1-0)i + (0-0)j + (0-0)k = so i take it this would equal a(1,0,0) which is a1?, so is a1.a2xa3 basically a1.a1 or am i horribly confused? it might be the latter but ill soldier on, if it is a1.a1 then does this not just give us the answer a(1,0,0)? or should i have got rid of the a at some point? or should i just be getting an integer? can someone please give me a gentle push in this question i really think i could do most of it myself im just a bit confused and stressed. But from here im stuck, i took 2 years off my degree and ive found that my basic vector calculation skills have left me completely. Do i do the cross product first or the dot? any help appreciated.