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The Angle Between Vectors - Problem 1

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.

So remember, a really neat application of the dot product, is that you can use it to find the angle between two vectors. Let’s do that here. We have vectors u equals <2, 6> and v equals <-1, 2>. Recall the formula; cosine of the angle between the vectors equals u.v, the dot product of two vectors, divided by the product of the magnitudes.

Let me calculate the magnitudes first, I usually like to get that out of the way. The magnitude of u is the square root of 2² or 4, plus 6² that’s 36. So root 4 plus 36, that’s root 40, which is 2 root 10. The magnitude of v is going to be the squared root of -1² or 1 plus 2² or 4. 1 plus 4, that’s root 5.

Cosine theta equals u dot v, 2 times -1, -2, plus 6 times 2, 12, over 2 root 10 times root 5. Now the numerator simplifies to 10. Over 2 root 10, root 5. Now let’s observe that there’s a little cancellation that I can do here. The root 10 cancels with the 10 leaving root 10 and the root 10 and the root 5 can cancel leaving root 2. So this really just equals root 2 over 2. What angle has a cosine that equals root 2 over 2? It’s 45 degrees. Don’t even need a calculator for that one. The angle between these two vectors is 45 degrees.

Compare that problem with the following. Where the vectors that are almost the same, it’s the same u vector but here I want to find the angle between u and the opposite of v. The opposite of the vector I just discussed. The angle between these guys is going to be u dot, and this vector is called negative v or the opposite of v, over the magnitude of u, times the magnitude of the opposite of v. The magnitude of u is still going to be 2 root 10. And the magnitude of the opposite of v, the opposite of v is the same length as v, just points me out to opposite direction. It will also be root 5. In the denominator, I’m still going to get 2 root 10 times root 5, but in the numerator I’ll get 2 times 1, 2, plus 6 times -2, -12. I get -10 over 2 root 10, root 5.

This is very similar to what I had before. I had 10 over 2 root 10 root 5. This is exactly the opposite. I’m going to have this simplified to negative root 2 over 2. What I have to figure out is what angle has a cosine equal to negative 2 over 2? It’s 135 degrees.

What’s interesting about this problem is, if you take the angle between 2 vectors, whatever answer you get for that, find the angle between the first vector and the opposite of the second, you’ll always get the supplement. That kind of makes sense. Here’s a little picture. If this is u and that’s v, 45 degrees, this is -v. It makes sense that this should be, 135 the supplement.

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