Hi Violina:
I want to suggest again you that use the cosine law, and calculate the distribution of the angle. It comes from choosing a particular orientation angle for v
1, like the North pole and allowing the other orientation angle to be any point on a unit sphere. Integrate over the longitude angle, φ, in polar coordinates (see the RHS of post #9 with r
2 dr=1) to get the distribution of the angle between v
1, the North pole, and v
2, which is α = π/2 - θ. Thus
cos α.= sin θ.
ADDED
I have had an insight following the digestion of DrClaud's posts, together with something I read in the Wikipedia article:
It can also be seen that the Maxwell–Boltzmann velocity distribution for the vector velocity
[vx, vy, vz] is the product of the distributions for each of the three directions:
Also, each of the three coordinate distributions, f(x), f(y) and f(z) are all the same Gaussian distribution. This means that my suggestion to use the Cosine Law and polar coordinates is a bad idea. Think about combining the factored form of the distribution for three coordinates, x, y, z, for two vectors. Think about the combined distribution for each coordinate separately.
Hope this helps.
Regards,
Buzz