Answer :
To determine which transformation is described by the rule [tex]\((x, y) \rightarrow (y, -x)\)[/tex], let's break down what this transformation represents.
### Step-by-Step Analysis:
1. Understanding the Initial Point:
- Let's start with a point [tex]\((x, y)\)[/tex] in the coordinate plane.
2. Applying the Transformation:
- According to the given rule, each point [tex]\((x, y)\)[/tex] is transformed into a new point [tex]\((y, -x)\)[/tex].
3. Visualizing the Transformation:
- To understand this transformation, let's take a few sample points and apply the transformation:
- Point [tex]\((1, 0)\)[/tex] transforms to [tex]\((0, -1)\)[/tex].
- Point [tex]\((0, 1)\)[/tex] transforms to [tex]\((1, 0)\)[/tex].
- Point [tex]\((1, 1)\)[/tex] transforms to [tex]\((1, -1)\)[/tex].
4. Recognizing the Rotation:
- We observe that moving a point [tex]\((x, y)\)[/tex] to [tex]\((y, -x)\)[/tex] corresponds to rotating the point 90 degrees counterclockwise about the origin.
- For example, starting at [tex]\((1, 0)\)[/tex] and rotating 90 degrees counterclockwise, we end up at [tex]\((0, -1)\)[/tex].
- Similarly, starting at [tex]\((0, 1)\)[/tex] and rotating 90 degrees counterclockwise, we end up at [tex]\((1, 0)\)[/tex].
5. Conclusion:
- Thus, the transformation [tex]\((x, y) \rightarrow (y, -x)\)[/tex] is the same as a 90-degree counterclockwise rotation about the origin.
### Selection of Correct Answer:
- Looking at the given options:
- [tex]\(R_{0,90^{\circ}}\)[/tex]
- [tex]\(R_{0,180^{\circ}}\)[/tex]
- [tex]\(R_{0,270^{\circ}}\)[/tex]
- [tex]\(R_{0,360^{\circ}}\)[/tex]
The correct transformation that matches [tex]\((x, y) \rightarrow (y, -x)\)[/tex], which rotates the point 90 degrees counterclockwise, is:
[tex]\[R_{0,90^{\circ}}\][/tex]
So, the answer is:
[tex]\[ \boxed{R_{0,90^{\circ}}} \][/tex]
### Step-by-Step Analysis:
1. Understanding the Initial Point:
- Let's start with a point [tex]\((x, y)\)[/tex] in the coordinate plane.
2. Applying the Transformation:
- According to the given rule, each point [tex]\((x, y)\)[/tex] is transformed into a new point [tex]\((y, -x)\)[/tex].
3. Visualizing the Transformation:
- To understand this transformation, let's take a few sample points and apply the transformation:
- Point [tex]\((1, 0)\)[/tex] transforms to [tex]\((0, -1)\)[/tex].
- Point [tex]\((0, 1)\)[/tex] transforms to [tex]\((1, 0)\)[/tex].
- Point [tex]\((1, 1)\)[/tex] transforms to [tex]\((1, -1)\)[/tex].
4. Recognizing the Rotation:
- We observe that moving a point [tex]\((x, y)\)[/tex] to [tex]\((y, -x)\)[/tex] corresponds to rotating the point 90 degrees counterclockwise about the origin.
- For example, starting at [tex]\((1, 0)\)[/tex] and rotating 90 degrees counterclockwise, we end up at [tex]\((0, -1)\)[/tex].
- Similarly, starting at [tex]\((0, 1)\)[/tex] and rotating 90 degrees counterclockwise, we end up at [tex]\((1, 0)\)[/tex].
5. Conclusion:
- Thus, the transformation [tex]\((x, y) \rightarrow (y, -x)\)[/tex] is the same as a 90-degree counterclockwise rotation about the origin.
### Selection of Correct Answer:
- Looking at the given options:
- [tex]\(R_{0,90^{\circ}}\)[/tex]
- [tex]\(R_{0,180^{\circ}}\)[/tex]
- [tex]\(R_{0,270^{\circ}}\)[/tex]
- [tex]\(R_{0,360^{\circ}}\)[/tex]
The correct transformation that matches [tex]\((x, y) \rightarrow (y, -x)\)[/tex], which rotates the point 90 degrees counterclockwise, is:
[tex]\[R_{0,90^{\circ}}\][/tex]
So, the answer is:
[tex]\[ \boxed{R_{0,90^{\circ}}} \][/tex]
