The right hand rule can be used to determine the direction
of the vector
resulting from any
application of the cross product
To use the magnetic portion of the Lorentz Force equation as an example:
F=q(U ⊗ B)
Then, positioning your hand as described above by Withnail, your thumb points in the direction of U, your index finger points in the direction of B, and your middle finger points in the direction of F.
Similarly, then, if we have any equation of the form:
Z=X ⊗ Y
Then X is your thumb, Y is your index finger, and Z is your middle finger.
In fact, the rectangular coordinate system most commonly used is defined by this very relation: use the equation above, and you can see that the positive x axis, the positive y axis, and the positive z axis follow this rule, where each axis is one of the vectors X Y Z, respectively. (In a right-handed coordinate system, of course.)
More generally, however, your thumb and index finger define a plane, to which your middle finger points perpendicular. This is to say that, while in many applications of the right hand rule the vectors represented by the index finger and thumb are perpendicular, they need not be at a right angle before you can successfully use the right hand rule.
Remember: DO NOT USE YOUR LEFT HAND. Trust me, it doesn't work too well to use the left hand rule when you need to use the right one (which is always, AFAIK), and you'd be surprised as to how easy it is to do this without thinking about it. ;^)
Also, there are two other ways to interpret your hand when it is in this position, both ways are also correct:
If your index finger is X, then your middle finger is Y, and your thumb is Z.
If your middle finger is X, then your thumb is Y, and your index finger is Z.
Here are a few (bad, sorry) diagrams that may help. Note that in each, X, Y, and Z all proceed counterclockwise around the diagram, and, in the diagram, the thumb always points to the right, the index finger always points up, and the middle finger always points toward you (with your palm facing you, of course):
index Y X Z
|_ thumb |_ X |_ Z |_ Y
/ / / /
middle Z Y X