The 'right hand rule' illustrates the Lorentz force law. This law describes the force exerted on a charged particle moving through a magnetic field. You can use your right hand to illustrate this law of physics as follows:

1. Extend you right hand as a fist but with the thumb sticking up, (the 'thumbs up hand sign').
2. Keeping thumb in position, point your index finger out, (like your hand is a pistol).
3. Keeping the 'cocked pistol' finger position, stick your middle finger out perpendicular to both your thumb and index finger.

Your thumb indicates the direction of motion of the charged particle.

Your index finger indicates the direction of the magnetic field.

Your middle finger indicates the direction of the force on the particle.

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

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
There is more than one right hand rule. To the best of my knowledge, there are actually several. The first one students encounter is the one that represents the Lorentz force law and is described very well above.

The second most common right hand rule is useful for determining the direction in which magnetic field lines propagate when a current is passed through a wire. To utilize this rule, loosely make a fist but stick your thumb completely out, like you're giving a "thumbs up" sign. Your thumb will represent the direction of the current's flow. Your fingers will dictate how the magnetic field lines curl around the wire, as if the field lines were originating at your knuckle and proceeding around towards your fingertips.

AwkwardSaw's definition of the right-hand rule (interchangeably known as the right-hand grip rule or right-hand screw rule in my physics class) can also be used to do the opposite: given a solenoid, determine where the north pole of a magnetic field is when current is run through the solenoid. Simply hold your hand out flat, curl your fingers around in the direction that the current is flowing, and extend your thumb. The north pole is at the end at which your thumb points.

The right-hand slap rule is another variant of the vector multiplication rule. Using the Lorentz force example, the thumb indicates direction of current (first vector), other extended fingers indicate direction of magnetic field (second vector), and the palm represents the direction of said Lorentz force (resultant vector). Why "slap"? Move your hand in the direction of the resultant vector, it makes a slapping motion.

Another method is, of course, the left-hand rule - exactly the same as the right-hand slap rule, except the first and second vectors are swapped. Oh, yeah, and while I'm here, I should mention this XKCD comic: it shows three more alternative rules for multiplication of vectors.

Funny no one mentions the key notion of laying the wire across your right palm, with thumb pointing where the current's going (i.e. towards battery cathode) and then curling and extending fingers as directed.  Much more visually evocative than simply grasping the air.  Be careful doing this with bare wire and/or in urban areas.

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