Math Problem Statement
what matrix represents a 180 degree rotation clockwise
Solution
A 180-degree rotation clockwise (or counterclockwise, since a 180-degree rotation in either direction results in the same transformation) around the origin in 2D can be represented by the following matrix:
-1 & 0 \\ 0 & -1 \end{pmatrix}$$ This matrix reflects the transformation that negates both the $$x$$ and $$y$$ coordinates of any point, effectively rotating it 180 degrees about the origin. Would you like more details or have any questions? ### Relative Questions: 1. How does a 90-degree rotation clockwise differ from a 180-degree rotation? 2. What would the matrix look like for a 270-degree rotation clockwise? 3. How does the determinant of the rotation matrix affect the area of a shape? 4. What are the new coordinates of a point $$(x, y)$$ after a 180-degree rotation? 5. Can a rotation matrix ever change the size of the shape it is transforming? ### Tip: For any angle $$\theta$$, the general rotation matrix is given by $$\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$$. For specific angles like 90°, 180°, and 270°, substitute the angle into the general formula to get the corresponding rotation matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Operations
Rotation Matrices
Formulas
General rotation matrix: \( \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \)
Theorems
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Suitable Grade Level
Advanced High School