So I was recently thinking about solar angles. A fairly straightforward, everyday example of solar angles.

 

Or so you'd think.

 

However, recently one thing came to mind. Suppose it was 6 hours before or after noon during a fall or spring equinox (for our purposes it wouldn't matter) at the equator. Since it was "halfway" between sun-over-the-horizon and sun-overhead, I presume the solar angle would be 45 degrees, right?

 

Now suppose at the same time someone else was, let's say, at 45 degrees latitude; (north or south for our purposes wouldn't matter) at the same time. How would one determine, then, what the solar angle there would be? Is there some sort of angular equivalent of the "vector components" used in physics and in linear algebra? If so, what would these angular equivalents be, and how would you add them to determine the combined effects of time of day and degrees of latitude on solar angle?

Yes - latitude and longitude are the angles.  In computing solar angle, one must include the effect of the tilt of the earth's axis.

So how would one deduce solar angle from difference in latitude/longitude from "wherever the sun is directly overhead?"

The question is vague.  I suspect that if you made it precise, you could answer it yourself.

Is there any mathematical formula (software notwithstanding) into which you could plug the latitude/longitude differences, from said position of direct sunlight, to get the solar angle, not unlike how you can use the pythagorean theorem to find a vector's magnitude from the magnitudes of its components?

4 hours ago, ScienceNostalgia101 said:

Is there any mathematical formula (software notwithstanding) into which you could plug the latitude/longitude differences, from said position of direct sunlight, to get the solar angle, not unlike how you can use the pythagorean theorem to find a vector's magnitude from the magnitudes of its components?

I think your investigation should begin with studying the spherical coordinate system: https://en.wikipedia.org/wiki/Spherical_coordinate_system

Understanding that will go a long way toward answering your questions.