# Anti-Commutative Law

1. Apr 26, 2007

### ND3G

Prove that anti-commutative law for the cross product: a * b = -(b * a)

The question looks easy enough except that I can not find a definition of the anti-commutative law anywhere.

I can find countless references to it but not a single definition. Can some one give me a run down on what the law entails?

Thanks

2. Apr 26, 2007

### matt grime

You wrote out the definition in the first line. * is anticommutative if a*b=-b*a for all a,b.

3. Apr 26, 2007

### nrqed

The law is directly stated in the question!!
It just says that if you reverse the order of the two vectors, you get minus one times the original result.

In genreal, consider an operation "F" that takes two quantities a and b as input and spits out a certain result. Let's write this as F[a,b] = R
(the result R could be a number, a vector, a matrix, whatever).
Is this function is antimmutative, it means that

F[b,a] = - F[a,b]

The dot product is commutative (switching the order of the two vectors multiplied gives the same result) whereas the cross product is anticommutative. Of course, a general function of two arguments does not have to be either commutative or anticommutative.