FCC Reciprocal to BCC: Proving Equation

[Piano man]

Homework Statement

The proof seems fairly straight forward, but after plugging in the primitive vectors to the equations for the reciprocal lattice vectors, I'm getting
$$2\pi\frac{\frac{a^2}{4}(x+y-z)}{\frac{a}{2}(y+z)\cdot(\frac{a^2}{4}(-x+y+z))}$$

One proof I checked said that the bottom line's
$$(y+z)\cdot(-x+y+z)=2$$

I don't see how. Any ideas?

 

[tiny-tim]

Hi Piano man!

If this is a dot product then x.y = y.z = z.x = 0.

 

[Piano man]

:facepalm:

thanks I completely forgot that.
So it would reduce to y.y + z.z = 2

A tad embarrassing not to spot that :D