When you cross two vectors, , you know that you will get a vector that is perpendicular to both and . If you study the illustration below, you will notice that and are pointing in different directions. To know exactly which direction is pointing, you use the right-hand rule. It goes like this:
Rule
You can use the right-hand rule to explain why the order actually matters in the cross product, because
This is because switching the order of and in the product is the same as switching the index finger with the middle finger. To do this, you have to turn your hand upside down, and then your thumb will point in the opposite direction, which gets you the negative of the original cross product.