Vectors have both a magnitude and a direction. We saw this in the last section. It should therefore be possible to write any vector as a product of its own magnitude times its own direction. If we had a vector called [math]\vec{a}[/math], it would look like this: [br][center][math]\vec{a}=|\vec{a}|\hat{a}=a\hat{a}.[/math][/center]The symbols are absolutely crucial to understand here. The arrow just describes a vector. The 'hat' describes a direction that points one unit in a particular direction. The 'a' describes the number of units to be taken in that direction. [b]It is common practice to write a vector quantity without its arrow and by doing so to imply the magnitude. This is shorthand that physicists use. [/b] In other words [math]a\equiv|\vec{a}|[/math] where the triple equals sign denotes equivalence.[br][br]We see the idea of writing a scalar times a unit vector in every vector we write. We described [math]\vec{r}=(3.0\hat{x}-4.0\hat{y})km.[/math] Each term independently is a scalar times a unit vector (with a hat on it). If your location was directly eastward from the college, then your position vector would have been just [math]\vec{r}=3.0\hat{x}km.[/math] This would mean that your location is 3.0 kilometer-sized steps in the direction of the x-axis. [br][br]In the actual example your location was 5.0 kilometer-sized steps in a direction -53 degrees from the x-axis. So the unit vector in general serves to indicate the direction like this one.[br][br]So what if I wanted to rewrite your location like [math]\vec{r}=r\hat{r}?[/math] How would I find the two terms? We already know [math]r=5.0km[/math] from our previous calculations. The direction[math]\hat{r}[/math] can be found by taking the vector and dividing by its magnitude, or [br][center][math]\hat{r}=\frac{\vec{r}}{r}.[/math][/center]Doing this with our vector gives [math]\hat{r}=\tfrac{3.0km\hat{x}-4.0km\hat{y}}{5.0km}=0.6\hat{x}-0.8\hat{y}.[/math] Note that the kilometers are now gone. In a strange twist of language,[b] a unit vector has no units in a dimensional sense[/b]. It should be clear that the magnitude of this unit vector is one. Not one kilometer. Just one. [b]The unit vector points in the specific direction of the vector[/b]. That's its whole job. So when asked for the direction of a vector, the correct response is to calculate the unit vector.

Vectors will be used all over the place in physics. Besides describing where things are, we need to describe how they are moving, how they are pushed, etc. Vectors will be used to describe these sorts of things as well as more abstract concepts like angular momentum and fields. [br][br]While vectors may seem like a burden now, they will really serve to make our lives much easier in the long run. I can't even imagine trying to do physics without vectors. It would be ugly, tedious and much more difficult if at all possible.