Matrices for coordinate rotations about a coordinate axis
  (here the coordinate system is being rotated)

Off[General :: spell] Off[General :: spell1]

Clear[xy2RotMat, z3RotMat, y3RotMat, x3RotMat] ; xy2RotMat[φ_] := (Cos[φ] Sin[ ... Cos[φ] Sin[φ] 0 -Sin[φ] Cos[φ]

Clear[vecLength] ; vecLength[x_] := (Flatten[x] . Flatten[x])^(1/2)

verticesOfTetrahedronDisplacementsForabcd =  {{0, 0, 1}, {0, (2 2^(1/2))/3, -1/3}, {-2/3^(1/2), -2^(1/2)/3, -1/3}, {2/3^(1/2), -2^(1/2)/3, -1/3}} ;

{aD, bD, cD, dD} = (verticesOfTetrahedronDisplacementsForabcd[[#1]] &)/@{1, 2, 3, 4}

{{0, 0, 1}, {0, (2 2^(1/2))/3, -1/3}, {-2/3^(1/2), -2^(1/2)/3, -1/3}, {2/3^(1/2), -2^(1/2)/3, -1/3}}

Some experimets:

vecLength[dD]

1

dD

{2/3^(1/2), -2^(1/2)/3, -1/3}

Ax = x3RotMat[-ArcTan[-2^(1/2)/3/-1/3]]

( ) 1 0 0 ... 2 1 Sqrt[-] ------- 0 3 Sqrt[3]

Ax = x3RotMat[-ArcTan[-1/3, -2^(1/2)/3]]

( ) 1 0 0 ... 2 1 -Sqrt[-] -------- 0 3 Sqrt[3]

Note that:

ArcTan[-1/3, -2^(1/2)/3]

-π + ArcTan[2^(1/2)]

but

ArcTan[-2^(1/2)/3/-1/3]

ArcTan[2^(1/2)]

Ay = y3RotMat[ArcTan[dD[[1]]/vecLength[{dD[[2]], dD[[3]]}]]]

( 1 2 ) ------- -Sqrt[-] ... 1 Sqrt[-] ------- 3 0 Sqrt[3]

Created by Mathematica  (February 18, 2004)