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.
Parametric equations are great for modelling motion, but they can also just be used to describe a curve. Consider the parametric equation; x equals 10 cosine t, y equals 6 sine t, for t between 0 and 2 pi. Let's eliminate the parameter to obtain a rectangular equation for the curve. That will help us identify what the shape is, so that we can graph it.
If x equals 10 cosine t, let's observe that x over 10 equals cosine t. If y 6 sine t, y over 6 equals sine t. Now one thing I know about sine and cosine, is that when you square them, and add the results together, you get 1 cosine squared, plus sine squared equals 1. If I substitute x over 10 for cosine, I get x² over 100. If I substitute y over 6 for sine, I get y² over 36, and that should look very familiar to you.
This is an ellipse, so the shape of the graph is going to be an ellipse. It's very helpful for when I'm going to actually graphing. It's an ellipse centered at the origin. Let's plot points, and then we'll be ready to graph the curve.
Now t goes between 0 and 2 pi, so let me just plot some easy points like 0, pi over 2, pi, 3 pi over, and 2 pi. Now cosine of 0 is 1, so 10 cosine 0 is going to be 10. Sine 0 is 0, so y will be 0. Let's observe that 2 pi has the same sine and cosine values, so we're going to get the same point when t equals 2 pi.
Now for pi over 2, cosine pi over 2 is 0, so this is 0. Sine of pi over 2 is 1, so I get 6. For pi, cosine is -1, I get -10 and sine is 0. For 3 pi over 2, cosine is 0, sine is -1 so I get -6. Let's plot these points. There's really only four points, these four. This one is the same as the first, but it's good to know that in the end, this particle or whatever we're plotting out comes back to the original point it started at.
Let's plot 10, 0. This is t equals 0, and then 0, 6. This is t equals pi over 2. Then -10, 0, that happens at t equals pi. Then 0, -6, that's t equals 3 pi over 2. Of course it returns back to this point at t equals 2 pi. It's an ellipse, so I don't need to plot anymore points. I can just draw the shape. It looks something like this, and that's it. If you want to indicate direction, you can throw in a couple of arrows here and there. The object starts here at 10, and then just orbits around the origin, and then comes back again, and stops.