To determine the correct way to state the transformation given by [tex]\((x, y) \rightarrow (y, -x)\)[/tex], we need to analyze the nature of the transformation.
1. The original coordinates [tex]\((x, y)\)[/tex] of a point are transformed to new coordinates [tex]\((y, -x)\)[/tex]. This type of transformation is a rotation because both coordinates change in a manner consistent with rotating around the origin.
2. Let's consider the specific effects of the transformation:
- The original x-coordinate becomes the new y-coordinate.
- The original y-coordinate changes sign and becomes the new x-coordinate, but with a negative sign.
3. To identify which rotation this corresponds to, visualize the point [tex]\((x, y)\)[/tex] on the coordinate plane:
- When we rotate a point 90 degrees counterclockwise around the origin, the new position of [tex]\((x, y)\)[/tex] would be precisely [tex]\((y, -x)\)[/tex].
4. This matches our transformation rule exactly. Therefore, the transformation [tex]\((x, y) \rightarrow (y, -x)\)[/tex] is indeed a rotation by 90 degrees counterclockwise around the origin.
Therefore, the correct way to state the transformation is [tex]\(R_{0,90^{\circ}}\)[/tex].
