To solve the problem of rotating a point [tex]\((x, y)\)[/tex] 90 degrees counterclockwise about the origin, we need to understand the transformation that occurs to the coordinates of the point during this rotation.
When we rotate a point [tex]\((x, y)\)[/tex] 90 degrees counterclockwise around the origin, the new coordinates [tex]\((x', y')\)[/tex] can be derived as follows:
1. The x-coordinate of the new point will be the negative of the original y-coordinate.
2. The y-coordinate of the new point will be the original x-coordinate.
The mathematical representation of this transformation is:
[tex]\[
(x', y') = (-y, x)
\][/tex]
Therefore, given a point [tex]\((x, y)\)[/tex], after a 90-degree counterclockwise rotation about the origin, the new coordinates will be [tex]\((-y, x)\)[/tex].
Example Calculation:
Suppose we have a point [tex]\((3, 4)\)[/tex]:
1. The new x-coordinate after rotation will be [tex]\(-4\)[/tex] (the negative of the original y-coordinate).
2. The new y-coordinate after rotation will be [tex]\(3\)[/tex] (the original x-coordinate).
So, the new coordinates of the point after rotation will be [tex]\((-4, 3)\)[/tex].
Hence, the function [tex]\(R(x, y) = (-y, x)\)[/tex] represents the point [tex]\((x, y)\)[/tex] rotated 90 degrees counterclockwise about the origin.
