[broken vacuum hose coming from back of engine, HELP]

thanks for all your help :/

[broken vacuum hose coming from back of engine, HELP]

I recently bought a 2001 golf vr6 and have NO IDEA WHAT I AM DOING. I understand basic car mechanics but this is my first vw so I guess this is also a gauge of the helpfulness of the community (nudge,nudge). Joking aside, I was ignorant in checking the car over before i bought it and now I have discovered that there is some kind of vacuum tube coming out of the back of the intake manifold that is disconnected on one side(the side going towards the intake drivers side) and the end going towards the manifold has broken at the nipple? Grommet? Idk what to call it but the piece that connects the hose to the manifold is broken and he tried to GLUE IT BACK. I guess it held as long as it needed to... those 2 tubes are connected by a t-fitting and the 3rd tube i have not ran down yet...

 

MY QUESTION is what is this tube? Where does the one tube that goes towards the drivers side intake actually need to be plugged in? and what is the piece that connects the bottom tube in the T to the manifold? part numbers or just an idea of what to look for would be nice. Thank you every one in advance, I will try to upload my pics of the 'T' but I am not very good at computers so good luck to me