p={1,5};  (*enter any point *)
angle=120 Degree  (* enter any angle *)

(* red is original point, green is rotated point *)

r=Norm[p];
plot1=ListPlot[{p},PlotStyle->{Red,PointSize[Large]},  PlotRange->{{-1.1*r,1.1*r},{-1.1*r,1.1*r}},  AspectRatio->1];
circle=Graphics[{Blue,Circle[{0,0},Norm[p]]}];
vector1=Graphics[{Red, Arrow[{{0,0},p}]}];

rotMatrix[t_]={{Cos[t],-Sin[t]},{Sin[t],Cos[t]}};
% //MatrixForm;

p2=rotMatrix[angle].p;

point2=Graphics[{PointSize[Large],Green,Point[p2]}];
vector2=Graphics[{Green, Arrow[{{0,0},p2}]}];
Show[plot1,circle, vector1, point2, vector2]