# [[Rotation by 90°, 180°, 270°]] (Mathematics > Geometry > Transformations) ## Definition Rotating a point around the origin involves moving it a certain angle counterclockwise (positive direction) or clockwise (negative direction). The most common rotations are 90°, 180°, and 270° counterclockwise. The general rotation formulas are as follows: - **90° Counterclockwise**: Swap the coordinates and negate the first: $ (x, y) \mapsto (-y, x) $ - **180° Counterclockwise**: Negate both coordinates (no swapping): $ (x, y) \mapsto (-x, -y) $ - **270° Counterclockwise** (or 90° Clockwise): Swap the coordinates and negate the second: $ (x, y) \mapsto (y, -x) $ ## Key Concepts - **Coordinate-Swap Rule**: 90° and 270° rotations involve swapping the coordinates, while 180° does not. - **Sign Changes**: Each rotation introduces specific sign changes: - 90°: First coordinate negated. - 180°: Both coordinates negated. - 270°: Second coordinate negated. - **Clockwise vs. Counterclockwise**: Clockwise rotations follow the same rules but in the reverse direction of their counterclockwise counterparts. ## Important Properties 1. **90° Counterclockwise Rotation**: $ (x, y) \mapsto (-y, x) $ 2. **180° Counterclockwise Rotation**: $ (x, y) \mapsto (-x, -y) $ 3. **270° Counterclockwise Rotation (90° Clockwise)**: $ (x, y) \mapsto (y, -x) $ ## Essential Formulas - **90° Counterclockwise**: $ (x, y) \mapsto (-y, x) $ - **180° Counterclockwise**: $ (x, y) \mapsto (-x, -y) $ - **270° Counterclockwise**: $ (x, y) \mapsto (y, -x) $ ## Core Examples 1. **90° Counterclockwise**: Rotate \( (3, 5) \) by 90°: $ (3, 5) \mapsto (-5, 3) $ 2. **180° Counterclockwise**: Rotate \( (-2, 6) \) by 180°: $ (-2, 6) \mapsto (2, -6) $ 3. **270° Counterclockwise (or 90° Clockwise)**: Rotate \( (4, -3) \) by 270° counterclockwise: $ (4, -3) \mapsto (-3, -4) $ ## Mnemonics to Remember: - **90° Counterclockwise**: Swap and negate the first coordinate: \( (x, y) \mapsto (-y, x) \) - **180° Counterclockwise**: Negate both coordinates: \( (x, y) \mapsto (-x, -y) \) - **270° Counterclockwise**: Swap and negate the second coordinate: \( (x, y) \mapsto (y, -x) \) ## Related Theorems/Rules - [[Rotation Matrices]]: Rotations can also be described using matrices. - [[Reflection Across Axes]]: Reflections across the x-axis or y-axis can be considered as special cases of rotations. ## Common Pitfalls - Confusing 90° counterclockwise with clockwise rotations. - Forgetting to swap the coordinates in 90° and 270° rotations. ## Related Topics - [[Rotation in a Plane]] - [[Rotation by 270 Degrees]] - [[Rotational Symmetry of Regular Polygons]] - [[Reflections of Functions]] - [[Reflection Across the Line y = -x]] - [[Translation]] - [[Effect of Multiplying by \(i\)]] - [[Rotation by Multiplying with \(i\)]] - [[Rotation by Multiplying with \\\(i\\\)]] - [[Rigid Motions]] - [[Rotation Matrices]] - [[Coordinate Transformations]] ## Quick Review Questions 1. What is the result of rotating \( (2, 7) \) by 180° counterclockwise? 2. How does a 270° counterclockwise rotation differ from a 90° clockwise rotation? *** ![[Rotation_by_90°,_180°,_270°_visualization]]