I am planning to make a simple biljardgame, once I have got a bit of time. But I want to make the collisions as realistic as possible. So how do I use vectors to calculate what the new routes will be after a collision?

I believe the book Physics for Game Developers has what you''re looking for.

But I can''t repeat the formula here. I don''t think it''s quite as simple as trading velocity components in the collision.

I found this searching the net, too:

Billiard Collision

There are one or two articles that deal specifically with this available at gamasutra.com.

this site is great:
http://www.d6.com/users/checker/dynamics.htm
check out the formula at the end of the third pdf, solves all your problems.