To determine the correct rule for rotating a point [tex]\( (x, y) \)[/tex] [tex]\( 90^{\circ} \)[/tex] about the origin, we need to consider how such a rotation affects the coordinates of a point in the coordinate plane.
When we rotate a point [tex]\( (x, y) \)[/tex] [tex]\( 90^{\circ} \)[/tex] counterclockwise about the origin, the new position of the point will have specific coordinates. The rotation causes a swap and change of signs in a predefined way.
Here is a step-by-step explanation of what happens during a [tex]\( 90^{\circ} \)[/tex] counterclockwise rotation:
1. The new x-coordinate becomes the negative of the original y-coordinate.
2. The new y-coordinate becomes the original x-coordinate.
Thus, a point [tex]\( (x, y) \)[/tex] will be transformed to [tex]\( (-y, x) \)[/tex].
Let's apply this to the given options:
- [tex]\( (x, y) \rightarrow (-x, -y) \)[/tex] is not correct because this corresponds to a [tex]\( 180^{\circ} \)[/tex] rotation.
- [tex]\( (x, y) \rightarrow (-y, x) \)[/tex] is correct because it matches our determined transformation for a [tex]\( 90^{\circ} \)[/tex] counterclockwise rotation.
- [tex]\( (x, y) \rightarrow (-y, -x) \)[/tex] is not correct because it does not correspond to a proper rotation matrix transformation for [tex]\( 90^{\circ} \)[/tex].
- [tex]\( (x, y) \rightarrow (y, -x) \)[/tex] is not correct because this corresponds to a [tex]\( 270^{\circ} \)[/tex] (or [tex]\( -90^{\circ} \)[/tex]) rotation.
Therefore, the correct rule that describes the transformation of a [tex]\( 90^{\circ} \)[/tex] rotation about the origin is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
So the answer is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
which corresponds to the second option in the list.
