I haven’t had the space yet to finish my Little 2021 A-to-Z, so let me resume playing the hits of past ones. For my first, Summer 2015, one, I picked all the topics myself. This one, Orthogonal, I remember as one of the challenging ones. The challenge was the question put in the first paragraph: why do we have this term, which is so nearly a synonym for “perpendicular”? I didn’t find an answer, then, or since. But I was able to think about how we use “orthogonal” and what it might do that “perpendicular ” doesn’t..
Orthogonal is another word for perpendicular. So why do we need another word for that?
It helps to think about why “perpendicular” is a useful way to organize things. For example, we can describe the directions to a place in terms of how far it is north-south and how far it is east-west, and talk about how fast it’s travelling in terms of its speed heading north or south and its speed heading east or west. We can separate the north-south motion from the east-west motion. If we’re lucky these motions separate entirely, and we turn a complicated two- or three-dimensional problem into two or three simpler problems. If they can’t be fully separated, they can often be largely separated. We turn a complicated problem into a set of simpler problems with a nice and easy part plus an annoying yet small hard part.
And this is why we like perpendicular directions. We can often turn a problem into several simpler ones describing each direction separately, or nearly so.
And now the amazing thing. We can separate these motions because the north-south and the east-west directions are at right angles to one another. But we can describe something that works like an angle between things that aren’t necessarily directions. For example, we can describe an angle between things like functions that have the same domain. And once we can describe the angle between two functions, we can describe functions that make right angles between each other.
This means we can describe functions as being perpendicular to one another. An example. On the domain of real numbers from -1 to 1, the function is perpendicular to the function . And when we want to study a more complicated function we can separate the part that’s in the “direction” of f(x) from the part that’s in the “direction” of g(x). We can treat functions, even functions we don’t know, as if they were locations in space. And we can study and even solve for the different parts of the function as if we were pinning down the north-south and the east-west movements of a thing.
So if we want to study, say, how heat flows through a body, we can work out a series of “direction” for functions, and work out the flow in each of those “directions”. These don’t have anything to do with left-right or up-down directions, but the concepts and the convenience is similar.
I’ve spoken about this in terms of functions. But we can define the “angle” between things for many kinds of mathematical structures. Once we can do that, we can have “perpendicular” pairs of things. I’ve spoken only about functions, but that’s because functions are more familiar than many of the mathematical structures that have orthogonality.
Ah, but why call it “orthogonal” rather than “perpendicular”? And I don’t know. The best I can work out is that it feels weird to speak of, say, the cosine function being “perpendicular” to the sine function when you can’t really say either is in any particular direction. “Orthogonal” seems to appeal less directly to physical intuition while still meaning something. But that’s my guess, rather than the verdict of a skilled etymologist.