ContourPlot3D[2 (x^2)+4(y^2)+(z^2)==45,{x,-10,10},{y,-10,10},{z,-10,10},MeshFunctions->{#1&,#2&,#3&},Mesh->{{{0}},{{0}},{{0}}},ContourStyle->{Opacity[0.4]},MeshStyle->{{Blue,Thick},{Blue,Thick},{Blue,Thick}},PlotPoints->20,Boxed->False]

Show[%,ContourPlot3D[4x-12y-z==45,{x,0,4},{y,-5,-1},{z,-3,1},Mesh->False,ContourStyle->{Opacity[0.4]},BoundaryStyle->{Thick,Black}],Graphics3D[{PointSize[Large],Black,Thick,Point[{2,-3,-1}],Arrow[{{2,-3,-1},{2,-3,-1}+(0.4){4,-12,-1}}]}]]

a=Show[ContourPlot3D[2 (x^2)+3(y^2)+(z^2)==9,{x,-4,4},{y,-4,4},{z,-4,4},MeshFunctions->{#1^2+#2^2-#3&},Mesh->{{{0}}},ContourStyle->{Opacity[0.3]},MeshStyle->{{Blue,Thick},{Blue,Thick},{Blue,Thick}},PlotPoints->20,Boxed->False],
ContourPlot3D[(x^2)+(y^2)-z==0,{x,-4,4},{y,-4,4},{z,-4,4},Mesh->False,ContourStyle->{Opacity[0.3]}]]

f[x_,y_,z_]=(x^2)+(y^2)-z;

g[x_,y_,z_]=2 (x^2)+3(y^2)+(z^2)-9;

gradf[x_,y_,z_]=Grad[f[x,y,z],{x,y,z}]

gradg[x_,y_,z_]=Grad[g[x,y,z],{x,y,z}]

{gradf[1,-1,2],gradg[1,-1,2]}

Cross[gradf[1,-1,2],gradg[1,-1,2]]

Join[{{i,j,k}},{gradf[1,-1,2],gradg[1,-1,2]}]//MatrixForm

TeXForm[%]

Show[a,ContourPlot3D[7x+6y+2z==5,{x,0.3,1.7},{y,-1.7,-0.3},{z,1.3,2.7},Mesh->False,ContourStyle->{Opacity[0.3]},BoundaryStyle->{Black,Thick}],Graphics3D[{PointSize[Large],Black,Thick,Point[{1,-1,2}],Arrow[{{1,-1,2},{1,-1,2}-(0.2){7,6,2}}]}]]