To rotate a point [tex]\((x, y)\)[/tex] [tex]\(90^{\circ}\)[/tex] counterclockwise about the origin, we can use a simple geometric transformation. Here's a detailed, step-by-step solution:
1. Understand the rotation transformation:
When you rotate a point [tex]\((x, y)\)[/tex] counterclockwise by [tex]\(90^{\circ}\)[/tex], the new coordinates [tex]\((x', y')\)[/tex] can be derived using the following transformation rules:
- The new x-coordinate will be the negative of the original y-coordinate, i.e., [tex]\(x' = -y\)[/tex].
- The new y-coordinate will be the original x-coordinate, i.e., [tex]\(y' = x\)[/tex].
2. Apply the transformation:
Given a point [tex]\((x, y)\)[/tex], the point after [tex]\(90^{\circ}\)[/tex] counterclockwise rotation will be:
[tex]\[
R(x, y) = (-y, x)
\][/tex]
3. Verification:
To verify this, consider how a point rotates around the origin:
- The point [tex]\((1, 0)\)[/tex] rotated [tex]\(90^{\circ}\)[/tex] counterclockwise becomes [tex]\((0, 1)\)[/tex].
- The point [tex]\((0, 1)\)[/tex] rotated [tex]\(90^{\circ}\)[/tex] counterclockwise becomes [tex]\((-1, 0)\)[/tex].
- The point [tex]\((-1, 0)\)[/tex] rotated [tex]\(90^{\circ}\)[/tex] counterclockwise becomes [tex]\((0, -1)\)[/tex].
- The point [tex]\((0, -1)\)[/tex] rotated [tex]\(90^{\circ}\)[/tex] counterclockwise becomes [tex]\((1, 0)\)[/tex].
These transformations match our derived rules.
4. Summary:
The function [tex]\(R(x, y)\)[/tex] that represents the point [tex]\((x, y)\)[/tex] rotated [tex]\(90^{\circ}\)[/tex] counterclockwise about the origin is:
[tex]\[
R(x, y) = (-y, x)
\][/tex]
This function can be easily implemented in various programming languages or used in mathematical applications whenever such a transformation is required.
