Let’s use some exercises with the dot product. Let u equal -4,5, v equals 3,6 and w equals 2,-5 a component form. Let’s take the dot product of u and v. Remember the way you calculate the dot product, is you multiply the first components of u and v in that order, and then you add to the second components of u and v.
So I’m going to multiply negative 4 and 3 plus 5 and 6. So here I get -12 plus 30. And that’s just 18. So U.V is 18. Now what about v.u? Here strictly speaking, you should multiply the components in the order that the product is written. So for V.U, I would multiply 3 times -4, the 3 comes first. And 6 times 5, the 6 comes first. But you are going to get the same result, because multiplication of real numbers is commutative. So you’ll get -12 plus 30, you’ll get 18.
V.W, well v is 3,6 and w is 2,-5. So it's going to be 3 times 2, plus 6 times -5. That’s going to be 6 minus 30 -24. What about W.V? I think we know what’s going to happen but let’s calculated anyway. Remember when you calculate W.V, the components of W come first. So it’s going to be 2 times 3 plus -5 times 6.
So I get 6 minus 30 again -24. So I think what this exercise demonstrates is that the dot product is commutative. Commutativity means that the order of the product doesn’t matter U.V, equals V.U.