We actually found the angle beta, not alpha, in today's examples.
If we had wanted alpha, we should have used arcsin, not arccos, 
in the final step of each of the two problems. 
I am sorry to say that I forgot to do this.

Since using arccos everywhere is easier to remember, 
just agree with me to call beta, not alpha, the angle between 
planes and lines.

In optics, beta is called the "angle of incidence":

https://en.wikipedia.org/wiki/Angle_of_incidence_%28optics%29

I've also heard beta called the "angle with the normal."

P.S. I'm going cook up some Python code to try to convince plotly.js
to create some accurate and, hopefully, interactive visualizations 
of today's two examples. I'll send you the results when/if I succeed.
