My geometric intuition is a bit weak, and I am trying to understand some basic examples of isometries in the Euclidean plane.
How can you write the following two isometries as a composition of first a function which can be either the reflection across the $x$-axis or the identity map, then a rotation about the origin, and finally a translation?
(i) The reflection across the line $y=3x$.
(2) The translation by $(2,2)$ followed by the reflection across $y=3x$.
I'd really appreciate an explanation.
For the first one, you start with a reflection on the $x$-axis and then you apply a clockwise rotation of $2\arctan(3)$ radians around the origin. Then you apply the null translation.
For the second one, start with a reflection on the $x$-axis and then you apply a clockwise rotation of $\arccos\left(-\frac45\right)$ radians around the origin. Then you apply the translation by $\left(-\frac25,\frac{14}5\right)$.