The Equation of a Plane

Friday December 28, 2018

I wasn't a great undergraduate student, but I had at least one great professor. She connected me with my post-graduate career path, and she taught some of my calculus classes. One day she turned briefly from the chalkboard and said with a smile:

"If you don't know equation of a plane, I kill you!"

I thought it was funny, and it wasn't what we were really working on in class, so I didn't think much more about it. But because I never thought it through then, doubt remained for some fifteen years: Was my life in danger?

That teacher wasn't even an associate professor back then, by official title. Recently, I learned that she's become an associate dean of the college. I was inspired to finally think about the equation of a plane.

Maybe she just meant that the equation of a plane is linear in Cartesian coordinates. That could have been it. But I think she was also getting at something about using the axis intercepts.

I learned $$y=mx+b$$ like a kind of incantation, and I don't think it was only because I went to Catholic schools. The "standard form" is treated like a purposeless afterthought, more about following rules than understanding anything.

But if the $$x$$-intercept is $$a$$ and the $$y$$-intercept is $$b$$, then $$bx + ay = ab$$ is a very nice equation of a line. Maybe call it both-intercepts form.

Adding a $$z$$-intercept of $$c$$, a plane then has a similar equation: $$bcx + acy + abz = abc$$.

I don't know for sure whether that's the equation that should have spared my life back then, but it's a charming one. It's fun to think about how it's related to other forms, and other dimensions—and how silly a rule like "$$A$$, $$B$$, and $$C$$ must be integers" is.

I wish I had been a better student back then. I had no idea how useful multivariate calculus and linear algebra would turn out to be, in particular. But I've been very lucky, and I'm honored to have had teachers who give me the pleasure of thinking things through, even little things like this, and even many years later than I should have. And I am deeply pleased to see that people who were so helpful to me have themselves found greater success, and positions from which to help many more people.