cil=ContourPlot3D[(y^2)+(z^2)==1,{x,-3/2,3/2},{y,-3/2,3/2},{z,-3/2,3/2},Mesh->False,Boxed->False];

plan=ContourPlot3D[(Sqrt[3])z+y==2,{x,-3/2,3/2},{y,-3/2,3/2},{z,-3/2,3/2},Boxed->False,ContourStyle->Opacity[0.2],Mesh->False];

{p,q}={{0,1/2,(Sqrt[3])/2},{1,1,(Sqrt[3])/3}};

Show[plan,cil,Graphics3D[{PointSize[Large],Thick,Point[{p,q}],Arrow[{p,q}]}]]

Clear[x,y,z]

u[x_,y_,z_]=(y^2)+(z^2)

Grad[u[x,y,z],{x,y,z}]

gradu[{x_,y_,z_}]=%

Show[plan,cil,Graphics3D[{PointSize[Large],Thick,Point[{p,q}],Arrow[{p,q}],Arrow[{p,p+0.4 gradu[p]}],Text["p",p+{0.1,-0.1,0}],Text["gradu[p]",p+0.4 gradu[p]+{-0.1,0,0.1}]}]]