Principal Component Analysis (PCA) vs Ordinary Least Squares (OLS): A Visual Explanation

Over at recently, a really interesting question was raised about principal component analysis (PCA). The gist was “Thanks to my college class I can do the math, but what does it MEAN?” I felt like this a number of times in my life. Many of my classes were focused on the technical implementations they kinda missed the section titled “Why I give a shit.” A perfect example was my Mathematics Principles of Economics class which taught me how to manually calculate a bordered Hessian but, for the life of me, I have no idea why I would ever want to calculate such a monster.