Question

Do Example 1 and Problem 3 if the transformation to a left-handed system is an inversion (see Problem 2).

Short answer

The expressions are proved below.

Step-by-step solution

Given information.

Matrix definitions are given.

Definition of a rotation matrix.

The rotation matrix is defined in this way.

$\left[\begin{array}{cc}\cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi \end{array}\right]$

Write the rotation matrices relation.

Write the rotation matrices about the three axes and defined them by an angle.

$\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \varphi & -\sin \varphi \\ 0& \sin \varphi & \cos \varphi \end{array}\right],\left[\begin{array}{ccc}\cos \varphi & 0& \sin \varphi \\ 0& 1& 0\\ -\sin \varphi & 0& \cos \varphi \end{array}\right],\left[\begin{array}{ccc}\cos \varphi & -\sin \varphi & 0\\ \sin \varphi & \cos \varphi & 0\\ 0& 0& 1\end{array}\right]$

Denote the three matrices by the rotation matrix symbols $R_x\left(f\right),R_y\left(f\right)\mathrm{and}{R}_z\left(f\right)$.

Denote the relations about y-z, x-z, x-y planes by the symbols ${\mathrm{O}}_{\mathrm{x}},{\mathrm{O}}_{\mathrm{y}}\mathrm{and}{\mathrm{O}}_{\mathrm{z}}$.

The element of this matrix is described by this formula.

${\left[O_k\right]}_{ij}={(-1)}^{{\delta }_{ik}}{\delta }_{ij}$

Continue evaluation.

$$\begin{gathered} {\mathrm{R}}_{\mathrm{k}}\left(\mathrm{\tau \tau }\right){\mathrm{O}}_{\mathrm{k}}={\mathrm{O}}_{\mathrm{k}}{\mathrm{R}}_{\mathrm{k}}\left(\mathrm{\tau \tau }\right) \\ =\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right] \\ =-\mathrm{I} \end{gathered}$$

Thus, an inversion is proved.

Prove it by using reflection and then rotation.

Define the rotation and reflection symbols.

$$\begin{gathered} {{\mathrm{R}}_{\mathrm{z}}}^{\text{'}}\left(\mathrm{\varphi }\right)={\mathrm{T}}^{-1}{\mathrm{R}}_{\mathrm{z}}\left(\mathrm{\varphi }\right)\mathrm{T} \\ {{\mathrm{O}}_{\mathrm{z}}}^{\text{'}}={\mathrm{T}}^{-1}{\mathrm{O}}_{\mathrm{z}}\mathrm{T} \end{gathered}$$

Combine rotation and reflection.

$$\begin{gathered} {{\mathrm{R}}_{\mathrm{z}}}^{\text{'}}\left(\mathrm{\tau \tau }\right){{\mathrm{O}}_{\mathrm{z}}}^{\text{'}}={\mathrm{T}}^{-1}{\mathrm{R}}_{\mathrm{z}}\left(\mathrm{\tau \tau }\right){\mathrm{TT}}^{-1}{\mathrm{O}}_{\mathrm{z}}\mathrm{T} \\ ={\mathrm{T}}^{-1}{\mathrm{R}}_{\mathrm{z}}\left(\mathrm{\tau \tau }\right){\mathrm{O}}_{\mathrm{z}}\mathrm{T} \\ =-{\mathrm{T}}^{-1}\mathrm{T} \\ =-\mathrm{l} \end{gathered}$$

Thus, an inversion is again proved.