Demonstration

Show[
Plot3D[-x/3-2y/3,{x,-5,5},{y,-5,5},Boxed->False,AxesOrigin->{0,0,0},BoxRatios->1,ViewPoint->{2,0.5,0.7}],
Graphics3D[{
Text["x", {5,0,0}],Text["y",{0,5,0}],Text["z",{0,0,5}],
Text[Style["u",Bold],{1.2,2.2,3.2}],
Green,Arrow[Tube[{{0,0,0},{1,2,3}},0.04]],
Purple,Arrow[Tube[{{0,0,0},{-2,1,0}},0.04]],Arrow[Tube[{{0,0,0},{-3,0,1}},0.04]]
}]
]

Worksheet Problem 1

Draw the vectors u and v:

Show[
ParametricPlot3D[{0,0,0},{t,0,1},Boxed->False,AxesOrigin->{0,0,0},BoxRatios->1,PlotRange->{{0,2},{0,2},{0,2}},ViewPoint->{2,0.5,0.7}],
Graphics3D[{
Text["x", {2,0,0}],Text["y",{0,2,0}],Text["z",{0,0,2}],
Text[Style["u",Bold,Darker[Green]],{1.1,1.1,1.1}],
Green,Arrow[Tube[{{0,0,0},{1,1,1}},0.02]],
Text[Style["v",Bold,Purple],{0,1.1,1.1}],
Purple,Arrow[Tube[{{0,0,0},{0,1,1}},0.02]]
}]
]

Draw the vectors u and v, and a basis vector for H:

Show[
ParametricPlot3D[{0,0,0},{t,0,1},Boxed->False,AxesOrigin->{0,0,0},BoxRatios->1,PlotRange->{{-2,2},{-2,2},{-2,2}},ViewPoint->{2,0.5,0.7}],
Graphics3D[{
Text["x", {2,0,0}],Text["y",{0,2,0}],Text["z",{0,0,2}],
Text[Style["u",Bold,Darker[Green]],{1.1,1.1,1.1}],
Green,Arrow[Tube[{{0,0,0},{1,1,1}},0.02]],
Text[Style["v",Bold,Purple],{0,1.1,1.1}],
Purple,Arrow[Tube[{{0,0,0},{0,1,1}},0.02]],
Red,Arrow[Tube[{{0,0,0},{0,-1,1}},0.02]]
}]
]

Draw the vectors u and v, and subspace H:

Show[
ParametricPlot3D[{0,-t,t},{t,-5,5},Boxed->False,PlotStyle->{Red,Dashed},AxesOrigin->{0,0,0},BoxRatios->1,PlotRange->{{-2,2},{-2,2},{-2,2}},ViewPoint->{2,0.5,0.7}],
Graphics3D[{
Text["x", {2,0,0}],Text["y",{0,2,0}],Text["z",{0,0,2}],
Text[Style["u",Bold,Darker[Green]],{1.1,1.1,1.1}],
Green,Arrow[Tube[{{0,0,0},{1,1,1}},0.02]],
Text[Style["v",Bold,Purple],{0,1.1,1.1}],
Purple,Arrow[Tube[{{0,0,0},{0,1,1}},0.02]],
Text[Style["H",Red],{0,-1.7,1.9}],
Red,Arrow[Tube[{{0,0,0},{0,-1,1}},0.02]]
}]
]

Diagram showing H and span(u,v), which are orthogonal complements.

Show[
ParametricPlot3D[{0,-t,t},{t,-5,5},Boxed->False,PlotStyle->{Red,Dashed},AxesOrigin->{0,0,0},BoxRatios->1,PlotRange->{{-2,2},{-2,2},{-2,2}},ViewPoint->{2,0.5,0.7}],
Plot3D[y,{x,-2,2},{y,-2,2},PlotStyle->{Opacity[0.6]}],
Graphics3D[{
Text["x", {2,0,0}],Text["y",{0,2,0}],Text["z",{0,0,2}],
Text[Style["u",Bold,Darker[Green]],{1.1,1.1,1.1}],
Green,Arrow[Tube[{{0,0,0},{1,1,1}},0.02]],
Text[Style["v",Bold,Purple],{0,1.1,1.1}],
Purple,Arrow[Tube[{{0,0,0},{0,1,1}},0.02]],
Red,Arrow[Tube[{{0,0,0},{0,-1,1}},0.02]]
}]
]