## Determining the Direction of a Cross Product

The direction of the cross product between two vectors is perpendicular to the plane the two vectors lie in. However, there are two ways this vector could point – essentially “up” or “down.” Conventionally, the choice between these two directions is determined by the “right hand rule.”

There are several contortions of your right hand that will tell you the direction of the cross product. My favorite is not the most well known, but I feel it is the easiest to remember. If you are calculating AxB, then you start with your hand open and flat, with your index finger parallel to A. Then, orient your wrist so that you can curl your fingers from the direction of A to that of B in the natural direction. Now, your thumb is pointing in the direction of the cross product.

The direction of the cross product differs depending on the order of the terms: AxB is not the same as BxA (in fact, these two cross products are negative multiples of each other).

As you can see, the unit vectors are all linked by cross products:

i x j = k

j x k = i

k x i = j

This relationship can be more concisely written as:
ea x eb=ec ε abc
where ea is as defined earlier and ε abc is equal to one when the numbers a b c are an “even permutation” of 1 2 3; negative one when the numbers a b c are an “odd permutation” of 1 2 3 and zero when a b c are not a permutation of 1 2 3 at all. This is often known as the “permutation tensor.” These relationships between the unit vectors are the definition of a Right Handed System. If you use the same relationships, but with the directions determined by your left hand in place of your right, you will have a :Left Handed System.