Co-rotational basis

(Subscript[(Underscript[(OverHat[E]), ~]), i]) are the laboratory Cartesian basis vectors and (Subscript[(Underscript[(OverHat[e]), ~]), i])((Underscript[t, ~]))
 are co-rotational Cartesian basis vectors

s1=;
With[{a=3,b=2,c=1},
ψ[τ_]:=τ;
θ[τ_]:=τ;
ϕ[τ_]:=τ;
Manipulate[
m=EulerMatrix[{ψ[τ],θ[τ],ϕ[τ]},{a,b,c}];
Show[{
Graphics3D[{Darker[Red],Arrow[{{0,0,0},{10,0,0}}]}],Graphics3D[{Darker[Red],Text["(Subscript[(Underscript[(OverHat[E]), ~]), 1])",{11,0,0}]}],
Graphics3D[{Darker[Blue],Arrow[{{0,0,0},{0,10,0}}]}],Graphics3D[{Darker[Blue],Text["(Subscript[(Underscript[(OverHat[E]), ~]), 2])",{0,11,0}]}],
Graphics3D[{Darker[Green],Arrow[{{0,0,0},{0,0,10}}]}],Graphics3D[{Darker[Green],Text["(Subscript[(Underscript[(OverHat[E]), ~]), 3])",{0,0,11}]}],
Map[TransformedRegion[#,AffineTransform[m]]&,{s1}],
Graphics3D[{Red,Arrow[{{0,0,0},m.{10,0,0}}]}],Graphics3D[{Red, Text["(Subscript[(Underscript[(OverHat[e]), ~]), 1])((Underscript[t, ~]))",m.{12,0,0}]}],
Graphics3D[{Blue,Arrow[{{0,0,0},m.{0,10,0}}]}],Graphics3D[{Blue, Text["(Subscript[(Underscript[(OverHat[e]), ~]), 2])((Underscript[t, ~]))",m.{0,12,0}]}],
Graphics3D[{Green,Arrow[{{0,0,0},m.{0,0,10}}]}],Graphics3D[{Green, Text["(Subscript[(Underscript[(OverHat[e]), ~]), 3])((Underscript[t, ~]))",m.{0,0,12}]}]
},
PlotRange->{{-10,10},{-10,10},{-10,10}}
],{τ,0,10}
]]