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I have a doubt in Chapter 5, section 2 (Group Motions of the Plane) of Michael Artin. Page 159 has a paragraph which describes how computations in group $M$ of all rigid motions in a plane (translation, rotation, and reflection) can be done using symbols as given in the attached picture. While it is easy to understand the compositions ticked in green, I have not understood the compositions boxed in red. References 2.3, 2.4 from the book also has been added to the picture.

Would be grateful for any help in understanding these.

enter image description here

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    $\begingroup$ Draw sketches for these compositions.. $\endgroup$ – Berci Mar 2 '18 at 8:13
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    $\begingroup$ To convince yourself that the transformations are the same, you can just multiply the relevant matrices $\endgroup$ – rbird Mar 2 '18 at 8:15
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I sketched a little diagram for the case

$$ \rho_\theta t_a = t_{a'}\rho_\theta $$

enter image description here

The parallelogram is just to show how $a' = \rho_\theta a$ is obtained: by rotating $a$. Using the same logic you can make sense of the other identity

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  • $\begingroup$ That was very helpful, thank you $\endgroup$ – SAK Mar 5 '18 at 16:43

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