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.
We’re talking about vector addition. Here I want to add the vectors OQ plus SQ, where my points O, Q and S are given by (0, 0) (-2, 1) and (4, 4). I’ve got these points plotted. OP plus SQ is what I want to draw. So let me first draw OQ, it goes from the origin to (-2, 1) and then SQ goes from point S to point Q looks like this.
Now this is not exactly the situation I want. Both of these vectors have their heads terminating at the same point. When you add vectors because you want to add them head to tail, so I need to translate one of these vectors. Let me translate the shorter of the two, I think that will be easier.I want to translate OQ so that it’s head is at the tail of SQ. OQ goes to the left 2 and up 1. I can put its initial point here, and then this will be an exact duplicate of OQ. That means OQ plus SQ will start from this point, and end at this point.
I’m just using the head to tail method to add these two vectors. This is the sum, starting at this point, ending at this point.
Now how long is this vector? What’s its magnitude? We’re asked to find that as well. Let me draw a right triangle; a horizontal leg and a vertical leg. Clearly the vertical leg is 2 units. How long is the horizontal leg? It’s 1, 2, 3, 4, 5, 6, 7, 8 units. The length of OQ plus SQ is the square root of 8² plus 2². That’s square root of 64 plus 4, root 68. What’s root 68? It’s 4 times 17 and that’s going to be 2 root 17.
Let’s do another one. RP plus PR. Let’s draw the sum of these two vectors and let’s find the length. RP is vector that starts at point R, and ends at point P, that’s this vector. Vector PR starts at point P and ends at point R. You’re probably noticing a couple of things. First of all, these vectors are already head to tail. Second of all, if I draw their sum, since they're both head to tail at both ends, I could start at the tail of RP, and end at the tail of PR. But those points are the same point. So my resultant or sum is a vector that starts at this point, and ends at this point.
This is the zero vector. It’s kind of hard to draw. It has no length. Because it has no length, I’m going to have to write that RP plus PR equals zero. The magnitude is zero, the length of their sum is zero. But it’s important that you know that there is such a thing as a zero vector. When you add two vectors which are opposite, you get the zero vector and its length is zero.