To determine which transformation corresponds to the rule [tex]\((x, y) \rightarrow (x, y)\)[/tex], let's analyze each option given, to understand their effects on points in the coordinate system.
1. [tex]\(R_{0,90^{\circ}}\)[/tex]: A rotation of [tex]\(90^{\circ}\)[/tex] counterclockwise around the origin. This transformation changes the coordinates [tex]\((x, y)\)[/tex] to [tex]\((-y, x)\)[/tex].
2. [tex]\(R_{0,180^{\circ}}\)[/tex]: A rotation of [tex]\(180^{\circ}\)[/tex] around the origin. This transformation changes the coordinates [tex]\((x, y)\)[/tex] to [tex]\((-x, -y)\)[/tex].
3. [tex]\(R_{0.270^{\circ}}\)[/tex]: A rotation of [tex]\(270^{\circ}\)[/tex] counterclockwise around the origin (or equivalently [tex]\(90^{\circ}\)[/tex] clockwise). This transformation changes the coordinates [tex]\((x, y)\)[/tex] to [tex]\((y, -x)\)[/tex].
4. [tex]\(R_{0,360^{\circ}}\)[/tex]: A rotation of [tex]\(360^{\circ}\)[/tex] around the origin. This transformation brings the point [tex]\((x, y)\)[/tex] back to its original position [tex]\((x, y)\)[/tex].
Given the rule [tex]\((x, y) \rightarrow (x, y)\)[/tex], it implies that the coordinates remain unchanged after the transformation.
Therefore, the transformation rule [tex]\((x, y) \rightarrow (x, y)\)[/tex] is equivalent to a rotation of [tex]\(360^{\circ}\)[/tex] around the origin. Thus, the correct notation for this transformation is:
[tex]\[ R_{0,360^{\circ}} \][/tex]
Hence, the answer is the transformation:
[tex]\[ R_{0,360^{\circ}} \][/tex]
which corresponds to the given result.
