Rotation Operator (vector Space) - Mathematical Formulation

Mathematical Formulation

Let

be a coordinate system fixed in the body that through a change in orientation is brought to the new directions

Any vector

rotating with the body is then brought to the new direction

i.e. this is a linear operator

The matrix of this operator relative to the coordinate system

$\begin{bmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{bmatrix} = \begin{bmatrix} \langle\hat e_1 | \mathbf{A}\hat e_1 \rangle & \langle\hat e_1 | \mathbf{A}\hat e_2 \rangle & \langle\hat e_1 | \mathbf{A}\hat e_3 \rangle \\ \langle\hat e_2 | \mathbf{A}\hat e_1 \rangle & \langle\hat e_2 | \mathbf{A}\hat e_2 \rangle & \langle\hat e_2 | \mathbf{A}\hat e_3 \rangle \\ \langle\hat e_3 | \mathbf{A}\hat e_1 \rangle & \langle\hat e_3 | \mathbf{A}\hat e_2 \rangle & \langle\hat e_3 | \mathbf{A}\hat e_3 \rangle \end{bmatrix}$

$\sum_{k=1}^3 A_{ki}A_{kj}= \langle \mathbf{A}\hat e_i | \mathbf{A}\hat e_j \rangle = \begin{cases} 0 & i\neq j, \\ 1 & i = j, \end{cases}$

or equivalently in matrix notation

$\begin{bmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{bmatrix}^T \begin{bmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

the matrix is orthogonal and as a "right hand" base vector system is re-orientated into another "right hand" system the determinant of this matrix has the value 1.

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