[One hour later I have gotten much closer] I have the following graphic which I wish to implement in a notebook

I get very close with this code

gr1 = Graphics3D[Arrow[{{0, 0, 0}, {1, 1, 1}}], Axes -> True, 
   Ticks -> False, Boxed -> False, 
   AxesLabel -> {"\!\(\*SubscriptBox[\(x\), \(1\)]\)", 
     "\!\(\*SubscriptBox[\(x\), \(2\)]\)", 
     "\!\(\*SubscriptBox[\(x\), \(3\)]\)"}, AxesOrigin -> {0, 0, 0}, 
   ViewPoint -> {2.628267, 2.026045, 0.6585882}, 
   Epilog -> {Rotate[
      Text[Style[Sqrt[Subscript[v, 1]^2 + Subscript[v, 2]^2], 
        FontSize -> 5], {.45, .16}], -45 Degree]}];
hp1 = HalfPlane[{{0, 0, 0}, {0, 0, 1}}, {1, 1, 0}];
pt = Graphics3D[{Blue, PointSize[.02], Point[{1, 1, 1}]}];
axis1 = Graphics3D[{Inset[
     "\!\(\*SubscriptBox[\(v\), \(1\)]\)", {1.1, .1, 0}], 
    Arrowheads[Medium], Thick, Arrow[{{0, 0, 0}, {1.5, 0, 0}}]}];
axis2 = Graphics3D[{Inset[
     "\!\(\*SubscriptBox[\(v\), \(2\)]\)", {.1, 1.1, 0}], 
    Arrowheads[Medium], Thick, Arrow[{{0, 0, 0}, {0, 1.5, 0}}]}];
axis3 = Graphics3D[{Inset[
     "\!\(\*SubscriptBox[\(v\), \(3\)]\)", {0, .1, .9}], 
    Arrowheads[Medium], Thick, Arrow[{{0, 0, 0}, {0, 0, 1.5}}]}];
ln1 = Graphics3D[{Red, Thick, Line[{{0, 0, 0}, {1, 1, 0}}]}, 
   Axes -> True];
ln2 = Graphics3D[{Red, Thick, Line[{{1, 1, 0}, {1, 1, 1}}]}, 
   Axes -> True];
ln3 = Graphics3D[{Red, Thick, Line[{{0, 0, 1}, {1, 1, 1}}]}, 
   Axes -> True];
ln4 = Graphics3D[{Red, Thick, Line[{{1, 0, 0}, {1, 1, 0}}]}, 
   Axes -> True];
ln5 = Graphics3D[{Red, Thick, Line[{{0, 1, 0}, {1, 1, 0}}]}, 
   Axes -> True];
Show[gr1, pt, axis1, axis2, axis3, ln1, ln2, ln3, ln4, ln5, 
 Graphics3D[{Opacity@0.2, hp1}], 
 ViewPoint -> {2.9684, 1.38468, 0.8469937}]

I am down to just two items: (1) getting the shading entirely inside the perimeter created by the vectors; and (2) making the little squares in the corners indicating 90 degree angles (Wolfram really needs a symbol for that).

Any help is (still) much appreciated.