Searching mathematical algorithm for vector calculation

Question

I have given directions of 3D Objects like this:

Direction1:

X-Vector:

X_X: 1

X_Y: 0

X_Z: 0

Y-Vector:

Y_X: 0

Y_Y: 1

Y_Z: 0

Z-Vector:

Z_X: 0

Z_Y: 0

Z_Z: 1

Direction2:

X-Vector:

X_X: 0

X_Y: 0

X_Z: 1

Y-Vector:

Y_X: 0

Y_Y: -1

Y_Z: 0

Z-Vector:

Z_X: 1

Z_Y: 0

Z_Z: 0

That looks like this (Direction1 on the left, Direction2 on the right):

I have to filter out the information about the rotation from direction1 to direction2 now. There are algorithms fe which calculates the rotation of vector1 to vector 2, but here i have 3 vectors and i don't know, how i can calculate the euler rotation angle here.

I thought about summarizing the 3 Vectors to 1, fe picture 1 would be (1,1,1) and pic2 would be (1,-1,1), but the problem here is that the information, which axe points in which direction gets lost. Has somebody an idea?

Answer 1

Seems that you want to find affine transformation that transforms one triplet of non-coplanar vectors into another triplet.
Make matrices A and B and unknown rotation matrix M .
Here column vector like x1 y1 z1 is your X_X X_Y X_Z and so on.

 M * A = B

    |x1 x2 x3 0|     |x1` x2` x3` 0|
M * |y1 y2 y3 0| =   |y1` y2` y3` 0|
    |z1 z2 z3 0|     |z1` z2` z3` 0| 
    |1  1  1  1|     |1   1   1   1| 

find inverse matrix InvA for the A and multiply both sides by IA

M * A * InvA = B * InvA
M * |1 |= B * InvA
M = B * InvA

Now you have matrix M needed to transform vectors.

Rotation about 5,0,0

      |1 0 0 -5|        |1 0 0 5|
M' =  |0 1 0  0| * M *  |0 1 0 0|
      |0 0 1  0|        |0 0 1 0|
      |0 0 0  1|        |0 0 0 1|