A Summer 2015 Mathematics A To Z: y-axis

y-axis.

It’s easy to tell where you are on a line. At least it is if you have a couple tools. One is a reference point. Another is the ability to say how far away things are. Then if you say something is a specific distance from the reference point you can pin down its location to one of at most two points. If we add to the distance some idea of direction we can pin that down to at most one point. Real numbers give us a good sense of distance. Positive and negative numbers fit the idea of orientation pretty well.

To tell where you are on a plane, though, that gets tricky. A reference point and a sense of how far things are help. Knowing something is a set distance from the reference point tells you something about its position. But there’s still an infinite number of possible places the thing could be, unless it’s at the reference point.

The classic way to solve this is to divide space into a couple directions. René Descartes made his name for himself — well, with many things. But one of them, in mathematics, was to describe the positions of things by components. One component describes how far something is in one direction from the reference point. The next component describes how far the thing is in another direction.

This sort of scheme we see as laying down axes. One, conventionally taken to be the horizontal or left-right axis, we call the x-axis. The other direction — one perpendicular, or orthogonal, to the x-axis — we call the y-axis. Usually this gets drawn as the vertical axis, the one running up and down the sheet of paper. That’s not required; it’s just convention.

We surely call it the x-axis in echo of the use of x as the name for a number whose value we don’t know right away. (That, too, is a convention Descartes gave us.) x carries with it connotations of the unknown, the sought-after, the mysterious thing to be understood. The next axis we name y because … well, that’s a letter near x and we don’t much need it for anything else, I suppose. If we need another direction yet, if we want something in space rather than a plane, then the third axis we dub the z-axis. It’s perpendicular to the x- and the y-axis directions.

These aren’t the only names for these directions, though. It’s common and often convenient to describe positions of things using vector notation. A vector describes the relative distance and orientation of things. It’s compact symbolically. It lets one think of the position of things as a single variable, a single concept. Then we can talk about a position being a certain distance in the direction of the x-axis plus a certain distance in the direction of the y-axis. And, if need be, plus some distance in the direction of the z-axis.

The direction of the x-axis is often written as $\hat{i}$, and the direction of the y-axis as $\hat{j}$. The direction of the z-axis if needed gets written $\hat{k}$. The circumflex there indicates two things. First is that the thing underneath it is a vector. Second is that it’s a vector one unit long. A vector might have any length, including zero. It’s convenient to make some mention when it’s a nice one unit long.

Another popular notation is to write the direction of the x-axis as the vector $\hat{e}_1$, and the y-axis as the vector $\hat{e}_2$, and so on. This method offers several advantages. One is that we can talk about the vector $\hat{e}_j$, that is, some particular direction without pinning down just which one. That’s the equivalent of writing “x” or “y” for a number we don’t want to commit ourselves to just yet. Another is that we can talk about axes going off in two, or three, or four, or more directions without having to pin down how many there are. And then we don’t have to think of what to call them. x- and y- and z-axes make sense. w-axis sounds a little odd but some might accept it. v-axis? u-axis? Nobody wants that, trust me.

Sometimes people start the numbering from $\hat{e}_0$ so that the y-axis is the direction $\hat{e}_1$. Usually it’s either clear from context or else it doesn’t matter.