After team “A” makes a basket, team “B” throws the ball the length of the court and out of bounds, without any body touching it, where do you administer the throw in? And where does it say it the rule book? This is a live ball correct? And should be back under teams “A” basket???

All throw-ins are live ball situations.

The ball comes back to the endline where the thrower stood, because the violation is by the thrower. 9-2-2