No, I don't think so...
But
EDIT: I'm not sure of that... yet.
EDIT2: Confirmed!
No, assuming .
Let , , and . Then, and .
If , then . The preceding equation cannot be solved; therefore, the answer is no.
I found a disproof on Wikipedia: Dot product - Wikipedia, the free encyclopedia
No, you are correct. My reasoning was flawed. Here is a disproof on Wikipedia: Dot product - Wikipedia, the free encyclopedia