Westonci.ca is the best place to get answers to your questions, provided by a community of experienced and knowledgeable experts. Connect with a community of experts ready to provide precise solutions to your questions quickly and accurately. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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.
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.
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.