Adding, subtracting,dot product etc of vectors when they never co-incide at any point

Question

Suppose I have a vectorA=3â+1ê at a point (1,2) and another vectorB=-3â at a point (-1,0) meaning that both vectors never co-incide at any point in 2D

Could the vectors then be added, subtracted, operated with a dot or cross product ?

Answer 1

Yes.

The definition of a vector do not consider the "base" point of where it emanates. So you generally just say that you have vectors $A=3\hat a+1\hat e$ and $B=-3\hat a$.

These may be added, subtracted etc.