There was many definitions of a vector.
The Classical Field Theory created physical vectors. For instance the electric field [intensity] in a point in space is E=grad(V).
The [electrical] potential V is a scalar which depends on its position in three-dimensional space.
Such vectors, arranged in a field of vectors in space may present function as divergence [scalar] and curl [vector] as field derivations.
Another definition of vector-phasor vector, or simply phasor-it represents a complex number-usually for current or voltage representation.
Since instantaneous value of current is i=sqrt(2)*I*cos(w*t+fi) [ where I it is the rms value and fi the angle for time=0] RE(I)=I*cos(fi) represents the actual value of current at time=0, divided by sqrt(2) . The imaginary part it is the virtual IM(I)=I*sin(fi) -minus for inductive load and plus for capacitive.
You may note is as I=I(cos(fi)+/-sin(fi)*i) and put it in a Cartesian two-dimension diagram. Here i=sqrt(-1) a non-existing entity. No gradient, divergence or curl is available here even no vectorial multiplication.
Now vector could be every thing: a matrix row or column, even a paper folder.
What is in a name?