Question

Explain why

Rω1 =R′ω1

Rω2 =R′ω2

Rω3 =R′ω3

for an Euler rotation.

Answer 1

Euler's theorem is that for every rotation R,there is a nonzero vector n for which Rn=n, that is n is an eigen vector of R associated with the eigen value 1.

A rotation matrix has the fundamental property that it's inverse is its transpose,that is

Where I is the 3*3 identity matrix and subscript T indicates the transposed matrix.Thst a rotation matrix has determinant +-1.

  

A rotation matrix with determinant +1 is a proper rotation and one with a negative determinant -1 that is a reflection combined with a proper rotation