I am trying to derive the formula for the volume of a Rhombohedron. I method is to find the three generating vectors and to then using the triple product to find the volume. According to the Wikipedia page on the Rhombohedron, the three generating vectors of a Rhombohedron with side length 1 are the following:
$\biggl(1, 0, 0\biggr),$\
$\biggl(\cos\theta, \sin\theta, 0\biggr),$
$\biggl(\cos\theta, {\cos\theta-\cos^2\theta\over \sin\theta}, {\sqrt{1-3\cos^2\theta+2\cos^3\theta} \over \sin\theta} \biggr).$
I was able to get the first two on my own, but I am unable to derive the third. Can anyone explain how to derive the third vector?