# [[Rotation by 270 Degrees]] (Mathematics > Geometry) ## Definition A **270-degree counterclockwise rotation** about the origin in the Cartesian plane is a transformation that rotates a point $(x, y)$ by $270^\circ$ counterclockwise, mapping it to a new point $(y, -x)$. This transformation is represented by the function: $ (x, y) \mapsto (y, -x). $ ## Key Concepts - **Counterclockwise Rotation**: A rotation in the positive direction around the origin. - **Rotation Matrix**: The transformation can also be represented using a rotation matrix: $ \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}. $ - **Origin-Centered Rotation**: The rotation occurs around the origin $(0, 0)$. ## Important Properties 1. **Distance Preservation**: The distance between points is preserved (isometry). 2. **Orientation**: The orientation of figures (e.g., triangles) is preserved but rotated. 3. **Specific Angle**: Rotating by $270^\circ$ counterclockwise is equivalent to rotating $90^\circ$ clockwise. ## Essential Formulas - To rotate a point $(x, y)$ by $270^\circ$ counterclockwise: $ (x, y) \mapsto (y, -x). $ - Rotation using a matrix: $ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} y \\ -x \end{pmatrix}. $ ## Core Examples 1. **Rotating point $A(3, 1)$ by $270^\circ$**: Using the formula, we compute: $ (3, 1) \mapsto (1, -3). $ The rotated point is $A'(1, -3)$. 2. **Rotating point $B(-2, 4)$ by $270^\circ$**: $ (-2, 4) \mapsto (4, 2). $ ## Related Theorems/Rules - **Rotation by $90^\circ$ counterclockwise**: $(x, y) \mapsto (-y, x)$. - **Rotation by $180^\circ$**: $(x, y) \mapsto (-x, -y)$. ## Common Pitfalls - Confusing counterclockwise rotation with clockwise rotation. - Incorrectly swapping or negating the coordinates in the transformation formula. ## Related Topics - [[Rotation Matrix]] - [[Transformations in the Plane]] - [[Isometries]] ## Quick Review Questions 1. What is the image of the point $(-4, 2)$ after a $270^\circ$ counterclockwise rotation? 2. How would you represent a $270^\circ$ rotation using a matrix? *** ![[Rotation_by_270_Degrees_visualization]]