To solve the given transformation [tex]\( (x, y) \rightarrow (x, y) \)[/tex], let's analyze what this transformation means and how it can be described geometrically.
This transformation [tex]\( (x, y) \rightarrow (x, y) \)[/tex] indicates that each point [tex]\((x, y)\)[/tex] is mapped to itself. In other words, every point in the plane remains in the same position after the transformation. This kind of transformation does not change any of the coordinates, so the positions of the points are exactly the same before and after the transformation.
Now, let's consider the given options, which represent rotations:
1. [tex]\( R_{0,90^{\circ}} \)[/tex]: This represents a rotation of 90 degrees about the origin.
2. [tex]\( R_{0,180^{\circ}} \)[/tex]: This represents a rotation of 180 degrees about the origin.
3. [tex]\( R_{0,270^{\circ}} \)[/tex]: This represents a rotation of 270 degrees about the origin.
4. [tex]\( R_{0,360^{\circ}} \)[/tex]: This represents a rotation of 360 degrees about the origin.
A rotation of 360 degrees brings every point back to its original position because it is a full circle rotation around the origin. Thus, the coordinates of every point remain the same after a 360-degree rotation.
Considering the given options, the correct representation of the transformation [tex]\( (x, y) \rightarrow (x, y) \)[/tex] is:
[tex]\[ R_{0,360^{\circ}} \][/tex]
So, the correct answer is [tex]\( R_{0,360^{\circ}} \)[/tex].
