What must they do to get the correct angle?
- They fire the LM's engine manually... usually, the computer operated the LM's engine. For the course correction, Jim Lovell operated it manually. There are two buttons in the LM fire control panel - one is START; one is STOP. He hit START, and he and Fred Haise manually controlled the ship while the engine was on. When it was time to end the burn, Jim hit STOP, and... that was that.
What is the reason they are diverted?
-Two reasons: 1 - The explosion of one oxygen tank rocked the craft, but there was jet of oxygen coming from Oxy tank 2.... it leaked for awhile - 2 hours, I think? - and this small jet caused the craft to alter it's attitude and trajectory, slightly.
and 2 - When the Oxy tank was finally empty, and they had powered up the LM, one of the LM's coolant processors vented spent refrigerant. Normally, sitting on the lunar surface, this is no problem... but, in space, it provided just a *little bit* of thrust, pushing the craft sideways, and off course. Not by much, but enough to where it was a problem.