The Dot Product of Vectors - Problem 2

We are talking about the dot product of vectors. Let U equal -4,5, v equal 3,6 and w equal 2,-5. Just for a review, let’s compute v plus w. Remember when you add two vectors, you add them component wise. So v plus w would be 3 plus 2, 5 and 6 plus -5, 1. So when I calculate U.V plus w, the v plus w is in parenthesis. That needs to be computed first, but I’ve just done that. So this is going to be U which is -4, 5.5,1.

Now the dot product of these two vectors is going to be -4 times 5, plus 5 times 1. And that’s -20 plus 5, -15. Now let’s compare that to this. What’s U.V? Let me calculate that first. U.V is going to be -4 times 3, -12, put those in parenthesis plus 5 times 6.

So plus, 30, plus U.W. U.W would be -4 times 2, -8 plus 5 times -5. -25. So we are going to get 18 plus -33. This is -15. You might have expected that because what this looks like is an expanded version of this. It looks like we distributed U over the v plus w. And that’s kind of what’s happened here although we don’t have a distributive property for the dot product yet. But this shows that the distributive property does work. That U does distribute over addition and it also happens to distribute over subtraction. So you can use that result in the future.