A vector that has a length of one is a unit vector.
We almost always choose unit vectors which are aligned with the axes of the system.
Examples:
- in 2D, we would probably choose unit vectors to be (0,1) and (1,0)
- in 3D, we would probably choose unit vectors to be (0,0,1), (0,1,0) and (1,0,0)
Appendix A
The hat symbol might be used to designate that a vector has a magnitude of 1.
Appendix B
If a set of unit vectors is correctly chosen, we can create any possible vector by multiplying our unit vectors by Scalars.
For example, the vector (3,4) can be obtained using unit vectors (0,1) and (1,0) and the scalars 3 and 4, as follows:
3(1,0) + 4(0,1) = (3,0) + (4,0) = (3,4)
We have the convenience that the values of the scalars match the values of the components of the vectors. There is a notion at work here that you will notice in future discussions of other topics.
Appendix A
It is important to know that we aren’t restricted to using (0,1) and (1,0) to form a Basis Set for creating vectors. The vectors in a legal basis set aren’t required to have lengths of 1.
Someone might come along and point out a counterexample where they use (0,2) and (2,0) and they can still get any vector. For this person, the scalars become 1.5 and 2:
1.5(2,0) + 2(0,2) = (3,0) + (0,4) = (3,4)
We also know that our choices for a Basis Set could be any vectors (a,b) and (c,d), provided that we choose (a,b) and (c,d) such that one is not proportional to the other. For example, we are not allowed to choose (1,2) and (2,4).
Vectoreverything 