To determine which rotation corresponds to the transformation rule [tex]\((x, y) \rightarrow (y, -x)\)[/tex], we need to examine how the coordinates change due to standard rotations about the origin.
1. 90-degree counterclockwise rotation [tex]\((R_{0, 90^{\circ}})\)[/tex]:
- Original point: [tex]\((x, y)\)[/tex]
- New point: [tex]\((-y, x)\)[/tex]
2. 180-degree counterclockwise rotation [tex]\((R_{0, 180^{\circ}})\)[/tex]:
- Original point: [tex]\((x, y)\)[/tex]
- New point: [tex]\((-x, -y)\)[/tex]
3. 270-degree counterclockwise rotation [tex]\((R_{0, 270^{\circ}})\)[/tex] (or 90-degree clockwise):
- Original point: [tex]\((x, y)\)[/tex]
- New point: [tex]\((y, -x)\)[/tex]
4. 360-degree counterclockwise rotation [tex]\((R_{0, 360^{\circ}})\)[/tex]:
- Original point: [tex]\((x, y)\)[/tex]
- New point: [tex]\((x, y)\)[/tex]
From the above information, the transformation [tex]\((x, y) \rightarrow (y, -x)\)[/tex] matches the new point's coordinates for a 270-degree counterclockwise rotation, or equivalently, a 90-degree clockwise rotation.
Thus, [tex]\((x, y) \rightarrow (y, -x)\)[/tex] corresponds to:
[tex]\[ R_{0,270^{\circ}} \][/tex]
So the correct answer is [tex]\( R_{0, 270^{\circ}} \)[/tex].
