To understand the transformation given by [tex]\((x, y) \rightarrow (y, -x)\)[/tex], we need to explore what this transformation does to a point in the Cartesian plane.
1. Original Point: Consider any point [tex]\((x, y)\)[/tex] in the plane.
2. Transformation: The rule [tex]\((x, y) \rightarrow (y, -x)\)[/tex] means the new coordinates of the point after the transformation will be [tex]\((y, -x)\)[/tex].
To determine which rotation this transformation represents, we can examine the effects of standard rotations around the origin:
- [tex]\(R_{0, 90^{\circ}}\)[/tex]: Rotates a point 90 degrees counterclockwise around the origin.
- The formula for this rotation is [tex]\((x, y) \rightarrow (-y, x)\)[/tex].
- [tex]\(R_{0, 180^{\circ}}\)[/tex]: Rotates a point 180 degrees counterclockwise around the origin.
- The formula for this rotation is [tex]\((x, y) \rightarrow (-x, -y)\)[/tex].
- [tex]\(R_{0, 270^{\circ}}\)[/tex]: Rotates a point 270 degrees counterclockwise around the origin or 90 degrees clockwise.
- The formula for this rotation is [tex]\((x, y) \rightarrow (y, -x)\)[/tex].
- [tex]\(R_{0, 360^{\circ}}\)[/tex]: This essentially brings the point back to its original position.
- The formula for this rotation is [tex]\((x, y) \rightarrow (x, y)\)[/tex].
Comparing the given rule transformation [tex]\( (x, y) \rightarrow (y, -x) \)[/tex] to these standard rotation formulas, we can see that [tex]\( (x, y) \rightarrow (y, -x) \)[/tex] matches the formula for [tex]\(R_{0, 270^{\circ}}\)[/tex].
Thus, the transformation [tex]\((x, y) \rightarrow (y, -x)\)[/tex] can be described as a rotation of 270 degrees counterclockwise around the origin.
Therefore, the correct answer is:
[tex]\(R_{0, 270^{\circ}}\)[/tex].
