Vectors fast approaches in Pre-Cal. It's probably my favorite topic for a lot of reasons. For instance, last year I got to experiment with visualizing 3D coordinates during vectors. It also served to confirm that my experiment with self-teaching wasn't the best idea. And lastly, I studied mechanical engineering and statics was like, totally my thing.

Years later I've realized that vectors are such a lovely demonstration of all the trig relationships we've studied AND let us discuss science. Students love to isolate math and science and groan at the thought that they could be best friends. The more science I can introduce into math, the better.

It also gives me an excuse to show them horrific videos of airplanes landing in crosswinds.

I've never had a good way to demonstrate the interplay of x and y components. It's usually involved a lot of hand waving and bad drawings of someone kicking a soccer ball. But, as always, Desmos to the rescue.

The black line denotes the vector with an adjustable magnitude. The x and y components trail along at the correct length at any angle you choose. The variable *c* operates in degrees, and the trig function of Desmos should be set to "degrees." Is it possible for Desmos to auto-detect when you want degrees? I'm not smart enough to know how you'd program that. I could launch into a rant about radians, but I won't.

The flexible points feature allows for a live read-out of the x and y component values, an easy way to let a student check their answer if I have them follow along somehow (there's a creative use for the polar grid here I think).

I hope this proves useful. I know someone is reading this knowing how to make a better version with parametric equations. I am so terrible at parametric equations.