One of the first places where people start getting confused in physics courses is the introduction of vectors. It doesn’t help that the material is always covered terribly. Some bearded blowhard in a plaid shirt stands at the front of the room and intones: “Vectors have both magnitude and direction”. He then starts doing a bunch of trigonometry problems. Any time you ask a question or make a mistake, the teacher declares: “No, that’s wrong! You forgot that vectors have magnitude and direction.”

We’re going to approach it from a different direction. Let’s start by going back to units. Units were a nice way of marking off numbers to make sure that we know what we’re talking about. When someone tell you to cook something in the microwave for fifteen feet, you know that that person is an idiot. We have units that describe how big things are (e.g. feet), how much things weigh (e.g. pounds), and how much time has passed (e.g. days). However, let’s say I wanted to know where something is?

Let’s say we meet on a small island and you’re looking for the bathroom. You ask if I know where one is and I say “One hundred feet way”. I’d get a well deserved punch in the face for that, because you still don’t know where the bathroom is. However, I can still be a jerk the other way around. I could point off in some direction and say “The bathroom is that way”, but not mention that it’s a ten mile hike to get there and you’d be better off squatting behind some bush.

If you really want to know where something is, you need both. I should point with my hand and say “five miles, that way”. There’s two parts to that phrase. The first part, where I say how far to go, is the called the magnitude. The second part, where I say which way to go, is the direction.

Now, it’s easy to write down “five miles”. I just wrote it. However, it’s a little harded to write down “that way”. The mathy way of writing a direction is the letter $e$ with a little hat. I don’t quite know what it’s the letter $e$: $\hat{e}$. If I come up with a good mnemonic device to remember that, I’ll make a comment here. Honestly, the letter itself doesn’t matter. In physics, I’ve seen $\hat{x}$, $\hat{i}$, and ever $\hat{q}$ at various points, so don’t get too caught up in the letters. Doesn’t really matter right now.

Getting back to the point, we can now write $\vec{b} = 5 \hat{e}$. We’ve given $b$ both a magnitude (5) and a direction ($\hat{e}$), so we say that $b$ is a vector and put a funny little arrow over the top, making $\vec{b}$.

Great, so now we’ve got these vectors. What do we do with them? I’ll save that for next week.