f[x_,y_,x0_,y0_] := ArcTan[(y - y0),(x-x0)];

x0=0; y0=0; Plot3D[f[x,y,x0,y0],{x,-1,1},{y,-1,1}]

single vortex:

VectorPlot[{Cos[f[x,y,0,0]],-Sin[f[x,y,0,0]]},{x,-3,3},{y,-3,3},VectorPoints->20]

single antivortex:

VectorPlot[{Cos[-f[x,y,0,0]],-Sin[-f[x,y,0,0]]},{x,-3,3},{y,-3,3},VectorPoints->20]

vortex-antivortex pair

VectorPlot[{Cos[f[x,y,0,0] - f[x,y,2,0] ],-Sin[f[x,y,0,0] - f[x,y,2,0]]},{x,-5,5},{y,-5,5},VectorPoints->20]