You might have noticed, but vectors only come in one type: straight lines. That's never going to change, but you can show more complex movement by making a vector field. To make a vector field, you'll need a function for the original vector. Most of the time, you'll be using unit vectors to do that. Unit vectors have a magnitude of 1 and a direction that's parallel to an axis, whether it's x, y, z or some an axis for some other direction we can't conceptualize.

 

A vector made up of its vector-y parts. The vector itself is the red line and it can be represented by its x-component (shown in blue) and its y-component (shown in orange).

When we're talking about applying unit vectors, the unit variables are going to be shown using the ̂ symbol, like this:

• x̂
• ŷ

A vector field is a function applied to the unit vectors. For example, the unit vector field for the function:

F̂(x, y) = -2yx̂ + 2xŷ

could look pretty mysterious, but we can figure out the shape by mapping some of the values of x and y.

In the first entry, we've started at the point (1,1), making it the starting point of the vector. The end point gets shifted two unit vectors to the left (because of that pesky negative sign) and 2 unit vectors up. Now all we need to do is draw an arrow starting at (1, 1) and pointing at (-1, 3). And, you know, do the same thing with the other three vectors. We got something like this in the end:

 

If we kept on going, we'd get a vector field with more and more red arrows rotating around the origin of the graph, but with wider and wider separations. Like a cyclone. Fancy.