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.
I want to talk a little bit about vector addition. Let’s draw each of these vectors and compute the magnitude starting with QR plus RP.
I have plotted points QR and P, they are (-2, 1) (-2, 6) and (4, 9). Let me draw QR plus RP. First QR. QR is this vector, vertical starting at Q, ending at R, looks like this. And then RP goes from here to here. These vectors are already head to tail. I can just draw their sum by starting at point Q and ending at point P. So let me draw that vector. This is their sum, their sum is QP.
We’re asked to find the length of this vector. The length of this vector, remember to find the length of a vector we have to draw a right triangle. So I have to draw a right triangle that goes from point Q to point P. I have to draw a horizontal leg and a vertical leg. Starting with the horizontal, and here is the vertical.
How long is the horizontal leg? It’s 1, 2, 3, 4, 5, 6 units. How long is the vertical length? 1, 2, 3, 4, 5, 6, 7, 8 units. The length of QR plus RP is the square root of 6² plus 8². That’s 36 plus 64, 100. Square root of 100, it’s 10.
Now remember in general you cannot find the length of the sum of two vectors, by finding the lengths of the two vectors and adding them. It’s not usually going to work that way, unless the vectors actually point in the same direction.
Let’s try another example. OS plus SR. We’ve got, let me erase some of these numbers, vector OS is this guy, from the origin to point S. And vector SR starts at point S and ends at point R, so it looks like this. We draw arrow heads here and I got too many vectors here. I’m going to get rid of this guy, I don’t need him. Get rid of this. Now the sum of the two vectors goes from the tail of the first to the head of the second. Let me draw that, vector OR.
We’re asked to find the sum of this vector, or rather the magnitude of this vector. We have to draw another right triangle. This won’t be too hard. Let me just draw a vertical leg right down to the horizontal axis and then my horizontal leg. So it’s two units across and 1, 2, 3, 4, 5, 6, units high. That means the length of OS plus SR is the square root of 2² plus 6². That’s the square root of 4 plus 36, root 40. Root 40 has a factor of 4, which is a perfect square so I pull that out and I get 2 root 10. That’s the length of the sum of vectors, 2 root 10.