How to find reciprocal lattice vectors

[girlinphysics]

So I know that the basis vectors of an FCC in a symmetric form are:
a = \frac{a}{2}(\hat{x} + \hat{y})
b = \frac{a}{2}(\hat{y} + \hat{z})
c = \frac{a}{2}(\hat{x} + \hat{z})

And that the reciprocal lattice vectors are the basis vectors of the BCC cells.
I'm having a hard time doing the cross products correctly, so if anyone could walk me through an example of the math that would be very helpful.

 

[girlinphysics]

Oh yes it was! I've edited it now, thanks.

 

[PRB147]

please read book by Kittle, introduction to solid state physics.