It is important to distinguish between true vectors and position vectors. A true vector does not depend on coordinate system but only on the difference between one end of the vector and the other. A position vector, in contrast, does depend on the coordinate system, because it is used to locate a position relative to a specified reference point. Some of the previous discussion may feel abstract to you. Do not worry about that because you gradually will become comfortable with the ideas as you use them and as you see then used in lectures.
Vector notation is simply a short-hand way of expressing information in a compact but precise way.
Answers to Activities 1.