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 are finding the parametric equations for line segments and here I have a problem. Let A equal to point 38, and B be the point 21, 16. I want the parametric equations for line segment AB.
Now remember the equations look like this, they're x equals and y equals. And you have 1 minus t, times the coordinates of the starting point; x1 y1. And I want A to be my starting point, so I’ll put 3 here and 8 down here. Plus t times the coordinates of the finishing point, the endpoint 21 and 16.
So these are my equations for 0 less than equal to t, less than equal to 1. Now let’s see what happens when we plug in: t equals 0, t equals 1 and t equals 1/2.
So for t equals 0, I’m going to get x equals 1 minus 0, 1, 3 plus 0 times 21, I get 3. When t equals 0, this term. 0 is out and I get 1 minus 0. 1 times 8. So I start off with the point 3,8. What about t equals 1? When t equals 1, these terms are going to disappear and I get x equals 1 times 21. x equals 21, and y equals 1 times 16, y equals 16.
So notice, that when t equals 0, I get my starting point. And when t equals 1, I get my endpoint. It is very important that you set up your equations this way with the 1 minus t, going with your starting point. Now what about t equals ½?
Here, when t equals ½, 1 minus t also equals ½. So I get ½ of 3, 3/2 plus ½ of 21, 21/2. That adds up to 24/2 which is 12. And I get ½ of 8 plus ½ of 16. That also adds up to 24/2 which is 12.
This point represents the midpoint of these two. Now let’s graph the parametric equations. Starting with point 3,8. So here is 3,8 that’s point A. And then point B is 21,16. 21, 5, 10, 15, 16 right here that’s point B. And I’ll show the midpoint for t equals 0; 12,12. So here’s 12, 5, 10, 12. That’s the midpoint I’ll call it m. So let me draw this.
Remember when you are graphing parametric equations, you should give some indications of what the t values are. Here I’ll put t equals 0, t equals 1 and for this point, t equals ½. If you like you can also put arrows indicating the direction of motion.
But if we were parametrizing an object moving, it would start at point A and it would end at point B. And half way through it would be at point m the midpoint.