Vectors and Parametric Equations
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.