Math Problem Statement
Determing the rotaion matrix for the following angle and direction theta = 150^deg counter clockwise
Solution
To determine the rotation matrix for an angle counterclockwise, we can use the standard 2D rotation matrix formula for counterclockwise rotation:
\begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}$$ ### Step 1: Find $$\cos(150^\circ)$$ and $$\sin(150^\circ)$$ - $$\cos(150^\circ) = -\frac{\sqrt{3}}{2}$$ - $$\sin(150^\circ) = \frac{1}{2}$$ ### Step 2: Plug these values into the rotation matrix: $$R(150^\circ) = \begin{bmatrix} -\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix}$$ This is the 2D rotation matrix for a counterclockwise rotation by $$150^\circ$$. Would you like further details or clarification on the steps? Here are 5 related questions to explore further: 1. How does the rotation matrix change for clockwise rotation? 2. What is the determinant of a 2D rotation matrix? 3. How do we apply this rotation matrix to a given vector? 4. What happens if the angle is negative? 5. How does the rotation matrix work in 3D? **Tip:** Always remember that for a rotation matrix, the determinant is always 1, ensuring that the rotation preserves the length of vectors.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Rotation Matrices
Formulas
2D Rotation Matrix
Theorems
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Suitable Grade Level
High School