Rotation Exercise

We are told that rotations from a U,V,W co-ordinate system are moved into an X(N),Y(E),Z system by the following matrix:

(({{n}, {e}, {z}}))=(1/(Sqrt[6]))(({{-2, 1, 1}, {0, (Sqrt[3]), -(Sqrt[3])}, {(Sqrt[2]), (Sqrt[2]), (Sqrt[2])}}))(({{u}, {v}, {w}}))  (Equation 1)
*We use lower-case letters (u,v,w,u,v,w) to denote the components of the vectors

In the drawing aboves, it is important to note that  U,V,W is a left-handed system and that X(N),Y(E),Z is a right-handed system.

In order to obtain the Galperin axes we have to perform rotations about the X,Y, and Z axes.  Several path permutations are avaibable.  We will try the following.  First, for example, a rotation matrix which moves lines (counterclockwise) by α radians around the X axis is :

U=RotationMatrix[α +Pi/2,{1,0,0}];
Um=RotationMatrix[α +Pi/2,{1,0,0}]//MatrixForm

Similarly, and in sequence, we use a  rotation matrix that moves lines by β radians counterclockwise around the Y axis:

V=RotationMatrix[β,{0,1,0}]
Vm=RotationMatrix[β,{0,1,0}] //MatrixForm

Combination of rotations: one about the X axis and the other about the Y axis, both in a counterclockwise direction produces a  matrix that shares similarities to the one that is cited as being used to perform these transformations (Equation 1).

Rotation=FullSimplify[U.V];
Rotationm = FullSimplify[U.V]//MatrixForm

An orthogonal matrix such as a rotation matrix above has an inverse that is also equal to its transpose, so that in order to to perform the inverse operation we have done above (moving X,Y, and Z to U,V, and W) we now move from axes U,V,W to N/X,E/Y,Z as follows:  (Equation 2)

invRotation = Transpose[Rotation] ;
invRotationm = Transpose[Rotation] //MatrixForm

If we consider that for Equation 1 we can mutliply the 1/(Sqrt[6]) throughout the matrix we can obtain the following new version of the same expression:

Solution = 1/Sqrt[6]{{-2,1,1},{0,Sqrt[3],-Sqrt[3]},{Sqrt[2],Sqrt[2],Sqrt[2]}}//MatrixForm

Then, in this format we see that the first component of the row of the matrix represents a counterclockwise rotation of 144.73 degrees (β), whose supplement is 35.3 degrees, the angle between the geophones and the horizontal plane.  We achive this result by comparing the component with the general rotation matrix (EQUATION 1)

beta = N[ArcCos[-(Sqrt[(2/3)])]/Pi * 180]

On the other hand, we also see that for the second component of the second row α= 135 degrees.  Note too that this is the supplement to 180 degrees.  This supplement (45 degrees) is the angle between each of the (U,V,W) axes and the vertical (Z)

alpha= N[ArcCos[-(Sqrt[(1/2)])]/Pi * 180]

Del[{a1,a2,a3}.{b1,b2,b3}]  //FullForm