Let's consider two right handed frames $\mathcal R_A$, with coordinate axes $i_A$, $j_A$ and $k_A$ and $\mathcal R_B$, with coordinate axes $i_B$, $j_B$ and $k_B$. Defining the rotation matrix $R^A_B$ as the change of basis matrix between base $A$ and base $B$, that is the orthogonal matrix whose columns are the components of unit vectors $i_B$, $j_B$ and $k_B$ with respect to frame $A$ $$R^A_B = \begin{pmatrix} i_B^\mathsf T i_A & j_B^\mathsf T i_A & k_B^\mathsf T i_A \\ i_B^\mathsf T j_A & j_B^\mathsf T j_A & k_B^\mathsf T j_A \\ i_B^\mathsf T k_A & j_B^\mathsf T k_A & k_B^\mathsf T k_A \end{pmatrix} $$ allows to change representation of a point between frame $B$ and frame $A$. For instance if $p^B$ is the position of a point represented in frame $B$, $p^A = R^A_B p^B$ is the position of the same point represented in frame $A$. This is the so called alias ("same") interpretation of the attitude matrix $R^A_B$.

On the other hand, the same matrix should represent also the rotation that transforms frame $A$ into frame $B$, that is $i_B = R^A_B i_A$, $j_B = R^A_B j_A$ and $k_B = R^A_B k_A$, according to the so called alibi ("elsewhere") interpretation of the same attitude matrix, because the vector is actually transformed into another vector by the rotation.

Which is the mathematic justification of these relations, given the definition of the matrix above? I tried to explain it by myself, observing that the matrix can be decomposed as $$R^A_B = \begin{pmatrix} i_A^\mathsf T \\ j_A^\mathsf T \\ k_A^\mathsf T \end{pmatrix} \begin{pmatrix} i_B & j_B & k_B \end{pmatrix} = \begin{pmatrix} i_A & j_A & k_A \end{pmatrix} ^ \mathsf T \begin{pmatrix} i_B & j_B & k_B \end{pmatrix} $$ equivalent to $$ \begin{pmatrix} i_A & j_A & k_A \end{pmatrix} R^A_B = \begin{pmatrix} i_B & j_B & k_B \end{pmatrix} $$ because all the matrices are orthogonal. In contrast, the equations $i_B = R^A_B i_A$, $j_B = R^A_B j_A$ and $k_B = R^A_B k_A$ might be gathered into $$ \begin{pmatrix} i_B & j_B & k_B \end{pmatrix} = R^A_B \begin{pmatrix} i_A & j_A & k_A \end{pmatrix} $$ which is a completely different equation, apparently. This aspect is often overlooked in 3D kinematics textbooks, that usually begin directly with the "change of representation point of view", taking for granted the "rotation point of view". Can someone give me a hint please?

  • $\begingroup$ As it seems, you have asked this also on math SE. It is unclear what kind of hint you are looking for. Perhaps, kinematic textbooks are concered with motion between frames, where you have the frames already defined. $\endgroup$ – 50k4 Jul 23 '18 at 12:18
  • $\begingroup$ I'm voting to close this question as off-topic because it is a duplicate of: math.stackexchange.com/questions/2845441/… $\endgroup$ – Ben Jul 23 '18 at 13:11
  • $\begingroup$ I have decided to post this question on this community, on purpose, because I thought it was the appropriate framework for such a kind of discussion. However, if it breaks the rules, do what you must. $\endgroup$ – Vexx23 Jul 23 '18 at 21:31

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