To determine the type of transformation represented by the rule [tex]\((x, y) \rightarrow (y, -x)\)[/tex], we need to identify the geometric change that the coordinates undergo.
1. Understanding the Transformation Rule:
- The given transformation rule is [tex]\((x, y) \rightarrow (y, -x)\)[/tex].
- This means that the point originally at [tex]\((x, y)\)[/tex] moves to a new location [tex]\((y, -x)\)[/tex].
2. Analyzing the New Coordinates:
- For a point [tex]\((x, y)\)[/tex], if we transform it according to the rule [tex]\((y, -x)\)[/tex], we observe:
- The [tex]\(x\)[/tex]-coordinate becomes the new [tex]\(y\)[/tex]-coordinate.
- The [tex]\(y\)[/tex]-coordinate becomes the negative of the new [tex]\(x\)[/tex]-coordinate.
3. Identifying the Rotation:
- In the coordinate plane, one common transformation that changes [tex]\((x, y)\)[/tex] to [tex]\((y, -x)\)[/tex] is a 90-degree counterclockwise rotation about the origin.
4. Verifying with Standard Rotation Matrices:
- A 90-degree counterclockwise rotation transformation can be written as:
[tex]\[
(x, y) \rightarrow (y, -x)
\][/tex]
- This matches the given rule exactly.
5. Confirming the Transformation Type:
- Therefore, we conclude that the transformation described by [tex]\((x, y) \rightarrow (y, -x)\)[/tex] is equivalent to a 90-degree counterclockwise rotation about the origin.
6. Final Answer:
- The transformation [tex]\((x, y) \rightarrow (y, -x)\)[/tex] is another way to state the rotation denoted by [tex]\( R_{0,90^{\circ}} \)[/tex].
Thus, the correct transformation is:
[tex]\[ \boxed{R_{0,90^{\circ}}} \][/tex]
