GIVE ANSWERS - Deutsch

Vergeblich is if you do something and u get nothing. e.g you gave your crush a diamond ring but it was "vergeblich" because he dont love you.

You can use "umsonst" the same way but its a bit worse language and you can use it aswell for free things (not paying). e.g Whats the price of these apples? They are for free (Sie sind umsonst)

"Nutzlos" is only a description for nouns like uselss.
e.g What is this roboter for? It's useless (er ist nutzlos) Or is your friend a good football player? No hes useless (Spielt dein Freund gut Fußball? Nein er ist nutzlos) u cant say nein er ist umsonst or nein er ist vergeblich. I hope I can help you