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
, it would look like this:
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.
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. In other words
where the triple equals sign denotes equivalence.
We see the idea of writing a scalar times a unit vector in every vector we write. We described
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
This would mean that your location is 3.0 kilometer-sized steps in the direction of the x-axis.
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.
So what if I wanted to rewrite your location like
How would I find the two terms? We already know
from our previous calculations. The direction
can be found by taking the vector and dividing by its magnitude, or
Doing this with our vector gives
Note that the kilometers are now gone. In a strange twist of language,
a unit vector has no units in a dimensional sense. It should be clear that the magnitude of this unit vector is one. Not one kilometer. Just one.
The unit vector points in the specific direction of the vector. That's its whole job. So when asked for the direction of a vector, the correct response is to calculate the unit vector.