activeX[
              u_] := {{1,  0,  0}, {0,  Cos[u],  -Sin [u]}, {0,  
                Sin [u],  Cos[u]}};
activeY[
              u_] := {{ Cos[u], 0,  Sin [u]}, {0,  1,  
                0}, {-Sin [u], 0,   Cos[u]}};
activeZ[
              u_] := {{Cos[u],  -Sin [u],  0}, {Sin [u],  Cos[u], 
                0 },  {0,  0,  1}};

Print["Active X:"]
MatrixForm[activeX[t]]
Print["Active Y:"]
MatrixForm[activeY[t]]
Print["Active Z:"]
MatrixForm[activeZ[t]]

Print["-----------------------------------"]
Print["Composite active rotation matrices about rotated \
coordinate frames"]
Print[""]
Print["Composite active rotation matrices transform the \
three coordinates of a vector"]
Print["from a fixed position to a rotated position in \
the same coordinate frame."]
\(Print["Coordinates of vector are in column matrices."]\)
Print["-----------------------------------"]

Print["Active rotatedXYX: \
activeX[a].activeY[b].activeX[c]"]
MatrixForm[
          activeX[a] . activeY[b] . activeX[c]]
Print["-----------------------------------\
"]

Print["Active rotatedXZX: \
activeX[a].activeZ[b].activeX[c]"]
MatrixForm[
          activeX[a] . activeZ[b] . activeX[c]]
Print["-----------------------------------\
"]

Print["Active rotatedYXY: \
activeY[a].activeX[b].activeY[c]"]
MatrixForm[
          activeY[a] . activeX[b] . activeY[c]]
Print["-----------------------------------\
"]

Print["Active rotatedYZY: \
activeY[a].activeZ[b].activeY[c]"]
MatrixForm[
          activeY[a] . activeZ[b] . activeY[c]]
Print["-----------------------------------\
"]

Print["Active rotatedZXZ: \
activeZ[a].activeX[b].activeZ[c]"]
MatrixForm[
          activeZ[a] . activeX[b] . activeZ[c]]
Print["-----------------------------------\
"]

Print["Active rotatedZYZ: \
activeZ[a].activeY[b].activeZ[c]"]
MatrixForm[
          activeZ[a] . activeY[b] . activeZ[c]]
Print["-----------------------------------\
"]

Print["Active rotatedXYZ: \
activeX[a].activeY[b].activeZ[c]"]
MatrixForm[
          activeX[a] . activeY[b] . activeZ[c]]
Print["-----------------------------------\
"]

Print["Active rotatedXZY: \
activeX[a].activeZ[b].activeY[c]"]
MatrixForm[
          activeX[a] . activeZ[b] . activeY[c]]
Print["-----------------------------------\
"]

Print["Active rotatedYXZ: \
activeY[a].activeX[b].activeZ[c]"]
MatrixForm[
          activeY[a] . activeX[b] . activeZ[c]]
Print["-----------------------------------\
"]

Print["Active rotatedYZX: \
activeY[a].activeZ[b].activeX[c]"]
MatrixForm[
          activeY[a] . activeZ[b] . activeX[c]]
Print["-----------------------------------\
"]

Print["Active rotatedZXY: \
activeZ[a].activeX[b].activeY[c]"]
MatrixForm[
          activeZ[a] . activeX[b] . activeY[c]]
Print["-----------------------------------\
"]

Print["Active rotatedZYX: \
activeZ[a].activeY[b].activeX[c]"]
MatrixForm[
          activeZ[a] . activeY[b] . activeX[c]]