How do I reconcile with a vindictive professor, or at least protect my grade?

I (foolishly) corrected my physics professor as he was about to teach fifty students a blatantly wrong simplification, and since then he's been out to get me. I get grades of 20% or lower on tests where I get all the answers right, but don't solve the problems in precisely the way he wants.

He answers all my questions with banal truisms ("Just use the provided equations and pick a decent reference frame and you'll get the right answer"), and even makes an effort to leave before collecting my assignments.

I'm afraid to complain, because he has a lot of political connections on campus, but I'm terrified that he's going to blemish my perfect GPA. Also, I can't withdraw from the class without falling behind in my coursework and needing to stay an extra semester.

What course of action would best protect my grades, and my future, from a vindictive professor?

Without knowing the professor, my bet is that you - unfortunately - will have little possibility mending your relationship with him. Maybe he is a believer in second chances, but judging from your description, I doubt it.

The first thing you need is a reality check - are you really being treated unfairly. You need to get a hold of some more senior students or friendly faculty members, it could either be at your own university or in an online community. Show them your graded coursework and the syllabus, and have them asses whether you are actually being treated unfairly. If you get down marked by 80 % or more, it should be obvious.

If you are, then your should contact your local dean immediately. Let him or her know the full story, it is their job to listen and deal with these things. Make it clear that you would like to just get this over with as easy as possible, but you don't feel safe taking the exam with this particular professor. If you are lucky, they have a procedure ready for these things. When you approach the dean, stay descriptive, and don't feel like you have to fight for your right. It should not matter, but unfortunately it does.

Make up your mind in advance whether or not you would be ok with transferring to another course. Even though it may slow you down a bit - or force you to read faster - it may be your best option.

When all this is done and over, just stay clear of that professor, and you should be ok. If I should guess, I would say that unless this is a very young professor, this is not the first occurrence of such a situation, and the professor will have a 'rumor', even among his political connections. You should be very unlucky if this comes back to bite you, and it would also be very unlikely that the professor would actually go out of his way to harm you.