QQuestionMathematics
QuestionMathematics
Based on what you know about rotations, what do you think the mapping rule is for a rotation of 270 degrees clockwise? Explain.
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Answer
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Step 1:I'll solve this step-by-step, focusing on the rotation transformation:
Step 2:: Understand the Rotation
A 270 -degree clockwise rotation is equivalent to rotating 3 / 4 of a full circle in the clockwise direction. This is the same as a - 90 -degree rotation.
Step 3:: Coordinate Transformation
- New y-coordinate: $$y_{new} = x
- New x-coordinate: x_{new} = -y
Step 4:: Algebraic Representation
\begin{bmatrix} x_{new} \ y_{new} \end{bmatrix} = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}
The rotation matrix for a 270 -degree clockwise rotation can be represented as:
Step 5:: Example Demonstration
- New y: $$y_{new} = x = 3
- New x: x_{new} = -y = - 4
Final Answer
The mapping rule for a 270 -degree clockwise rotation is (x, y) \rightarrow (-y, x), which transforms coordinates by negating the y-coordinate and setting the new y to the original x-coordinate.
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