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The Dot Product of Vectors - Problem 2

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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.

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