To reuse an old, bad, joke in a new context - given carefully controlled experimental conditions, when conducting precisely controlled and well thought out scientific experiments on women, one finds that they do what they damn well please :-)
As far as MRI goes, you've gotten me curious about the topic, so I've been looking into it. There are some rather subtle things going on. The magnetic dipole moment, though, is a farily easy part of the situation to understand. (The author ducks an imagined glare from Tsunmai). No, really, that part isn't that hard.
What you have are a bunch of bar magnets, in essence. And you put them in a magnetic field. Just like electric dipoles, they tend to align with the field. So, obviously, when you turn on the magnet in an MRI machine, all the protons just line up to be parallel with the field, right?
Ummm - ooops, no that's not right. OK, well, we ignored temperature, and these protons are all moving around and being jostled, so only SOME of them line up. Right?
<read read> Hmmm, it says here that the precess.
Oh yeah, that's right, the protons are all like little gyroscopes too. OK, I can see why the might precess if they weren't lined up - it's the same as any other gyroscope, when you put a torque on it, it precesses. Going back to our bar magnets, that's just what the magnetic field does, it trys to "twist" them back into position.
The author visualizes one of the toy gyroscopes he played with as a kid. If it's spinning rapidly, it balances on its point nicely. As the spin winds down, the gyroscope starts to wobble, or precess. The tip of the gyroscope moves in a circle. This can be explained quite nicely mathematically. You usually get a full treatment in an introductory graduate physics course, like Goldstein's "Classical Mechanics". The author imagines Tsunami's probable reaction to graduate level physics, and decides that it would be a good idea to skip a rigorous treatment of gyroscopes. This also saves him from having to re-read the textbook
. l guess all we really need to know is that it's torque that makes a gyroscope precess. And we can see that the magnetic field provides a torque on the magnetic dipoles. So that part all makes sense.
What's a little difficult to see, though, is what controls the angle of precession, or why it has to happen at all. It's fairly easy to imagine the behavior of a bunch of classical bar magnets, a little trickier to imagine the behavior of a bunch of rotating gyroscopic bar magnets. But now we have quantum-mechanical rotating bar magnets...
As far as the energy, goes, though, it's easy enough to see that the total energy will be the dot product of the magnetic field and the vector representing the magnetic dipole.
. This just means that when the two vectors point in the same direction, one multiplies their lengths, making the total energy equal to the field strength times the dipole strength when the dipole is perfectly aligned.
The energy varies smoothly as one changes the angle, droping to zero when the dipole is oriented perpendicularly to the field. Here the energy is zero, but the torque is a maximum. When one anti-aligns, the energy is the lowest, being the negative of the energy in the first case.
The rule for dot products can be written as \vec{a} \cdot \vec{b} = |a||b| cos(\theta), where \theta is the angle between the vectors, which is 0 when they point in the same direction, and 180 when they point in opposite directions.
The point of calculating the energy is to know what the Lamour frequency will be. This again comes from quantum mechanics, it's the famous relation between the energy of a photon , and it's frequency.
But we don't actually have to get bogged down in these details of how the Lamour frequency is actually calculated to get some idea of how the system works. All we need to know is that the Lamour frequency depends on the magnetic field strength. Thus varying this strength varies the frequency, and allows one to distinguish position. It's not quite clear yet how one determines all of x,y, and z via this mechanism though (it would seem like there are too many variables to encode with one magnetic field strength).
But it was an interesting question! I wanted an answer and I wanted it fast! 
Why am I even ASKING this?? I'm not going to understand the dang answer anyway... )
Well. It looks like seizure time again. 
I suppose you just have to Take It On Faith. When you multiply a vector by a number, you keep the direction of the vector unchanged, but you multiply it's length by said number.
