You may have noticed that using the Cartesian representation to represent displacement vectors (i.e. ) is a little inconvenient. One shortcoming they have is that it is not immediately clear how far one moves when undergoing the displacement . A quick calculation makes it clear that we move a distance , suggesting that our displacement (and all its vector representations) have the length . But now we have the distance travelled in the length as well as in the direction suggested by .
A common way to ease notation is to explicitly break vectors up into a scalar length and a direction vector such that all direction vectors have the same length, 1. We say that these are unit vectors (because they have a length of one unit), in the direction that they point. For example, the displacement given by has length in the direction .
We can then say that and it is clear what we mean. A movement of length in the direction . Here is a number, and is inherently independent of the coordinate system. too is independent of coordinates, but its specific representation will depend on the system we use (cylindrical, polar, Cartesian, et cetera).
If we want to repeat the displacement two times, we can make the new displacement . This repetition is not limited to integer numbers, we can have , , or in general that represents a continuous motion in time.
As we explore more topics in mechanics, electrodynamics, relativity, and beyond, the extension and contraction of vectors along their direction will be a fundamental tool.