Now stuff has become so easy ...
In[3]:= Sin[VectorAngle[{2, -1, 1}, {3, 2, 1}]]
Out[3]= Sqrt[59/21]/2
VectorAngle and the formula is of course
In[4]:= Sin[ArcCos[Dot[{2, -1, 1}, {3, 2, 1}]/(Norm[{2, -1, 1}] Norm[{3, 2, 1}])]]
Out[4]= Sqrt[59/21]/2
as the help spells it out.
