I need the help of competent specialists in linear algebra in Mathematica.
In the Euclidean space, two pyramids rotate at different angles. Its vertices coincide with the beginning of the base coordinate system at the point (0,0,0). The coordinates of the remaining vertices are described by a triple of threedimensional vectors corresponding to each pyramid:
$ W_ {11}, W_ {12}, W_ {13} $ – for the first pyramid;
$ V_ {21}, V_ {22}, V_ {23} $ – for the second pyramid, we need to find these vectors;
The coordinates of the vectors for the first pyramid are known. It is necessary to find the coordinates of the three remaining vectors for the second pyramid. In this case, two conditions must be met:

Scalar product $ W_ {1j} cdot W_ {1j} $, $ W_ {2j} cdot W_ {2j} $ Y $ W_ {3j} cdot W_ {3j} $, where $ j = $ 1,2.3 of two corresponding vectors taken for the first and second matrix must be the alpha angle.

The scalar product of two adjacent vectors for the second pyramid must be the betta angle.
In the form of equations, these conditions can be written as:
begin {cases} W_ {11} cdot V_ {21} = cos ( alpha) \ W_ {12} cdot V_ {22} = cos ( alpha) \ W_ {13} cdot V_ {23 } = cos ( alpha) \ V_ {21} cdot V_ {22} = cos ( beta) \ V_ {22} cdot V_ {23} = cos ( beta) \ V_ {21} cdot V_ {23} = cos ( beta) end {cases}
I cannot solve this system of equations directly and find the vectors I need, and I am not sure if this is possible using the vector array apparatus.
Numerically, this problem was not resolved in Mathematica; apparently, too complex equations are obtained. For this, I added unit length conditions for the vectors of the second pyramid to the equations.
In(5):= pars = {Subscript((Alpha), 2) = 95*Pi/180,
Subscript((Alpha), 3) = 95*Pi/180,
Subscript(w, 1) = {0.2, 0.8, 0.3},
Subscript(w, 2) = {0.4, 0.9, 0.1},
Subscript(w, 3) = {0.2, 0.25, 0.38},
Subscript(v, 1) = {Subscript(v, 11), Subscript(v, 12),
Subscript(v, 13)},
Subscript(v, 2) = {Subscript(v, 21), Subscript(v, 22),
Subscript(v, 23)},
Subscript(v, 3) = {Subscript(v, 31), Subscript(v, 32),
Subscript(v, 33)}};
}
NSolve({Dot(Subscript(w, 1), Subscript(v, 1)) ==
Cos(Subscript((Alpha), 2)),
Dot(Subscript(w, 2), Subscript(v, 2)) ==
Cos(Subscript((Alpha), 2)),
Dot(Subscript(w, 3), Subscript(v, 3)) ==
Cos(Subscript((Alpha), 2)),
Dot(Subscript(v, 1), Subscript(v, 2)) ==
Cos(Subscript((Alpha), 3)),
Dot(Subscript(v, 2), Subscript(v, 3)) ==
Cos(Subscript((Alpha), 3)),
Dot(Subscript(v, 1), Subscript(v, 3)) ==
Cos(Subscript((Alpha), 3)), Norm({Subscript(v, 1)}, 2) == 1,
Norm({Subscript(v, 2)}, 2) == 1,
Norm({Subscript(v, 3)}, 2) == 1}, {Subscript(v, 11),
Subscript(v, 12), Subscript(v, 13), Subscript(v, 21),
Subscript(v, 22), Subscript(v, 23), Subscript(v, 31),
Subscript(v, 32), Subscript(v, 33)})