Final answer:
The result of a 270-degree counterclockwise rotation of a point (x, y) in the Cartesian plane is (-x, -y), achieved by a 90-degree clockwise rotation rule followed by sign changes for the counterclockwise direction.
Step-by-step explanation:
The result of a 270-degree counterclockwise rotation of a point in the Cartesian plane can be determined by applying rotation rules. Starting with any point (x, y) in the plane, a 270-degree counterclockwise rotation is equivalent to a 90-degree clockwise rotation, since completing a full 360-degree rotation would bring the point back to its original position. To execute this rotation, we can rotate the point 90 degrees clockwise, which is easier to visualize, and then simply adjust to the counterclockwise direction by making a sign change.
Here is a step-by-step explanation of how this rotation works:
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- Identify the original point coordinates (x, y).
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- Apply the 90-degree clockwise rotation rule, which swaps the coordinates and changes the sign of the new y-coordinate, resulting in (y, -x).
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- Adjust the sign for a counterclockwise direction by converting (y, -x) to (-x, -y), which is the result of a 270-degree counterclockwise rotation.
Therefore, the new position of the point after a 270-degree counterclockwise rotation is (-x, -y). This rotation is commonly used in various mathematical applications as well as in computer graphics, robotics, and physics concepts like torque and angular velocity, as seen in the provided references.