3π The matrix that rotates by- is 4 X - 1/2 √2-1/2 √2 12 -1/√2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Title: Rotation Matrix for \( \frac{3\pi}{4} \) Angle**
**Description:**
The task is to identify the matrix that performs a rotation by an angle of \( \frac{3\pi}{4} \).
**Matrix:**
\[
\begin{bmatrix}
-\frac{1}{2}\sqrt{2} & -\frac{1}{2}\sqrt{2} \\
\frac{1}{2} & -\frac{1}{2}\sqrt{2}
\end{bmatrix}
\]
**Explanation:**
This image shows a 2x2 matrix surrounded by a border indicating it was selected, but marked with an 'x', suggesting the selection is incorrect. For a rotation by \( \frac{3\pi}{4} \), the correct matrix is typically derived from:
\[
\begin{bmatrix}
\cos\left(\frac{3\pi}{4}\right) & -\sin\left(\frac{3\pi}{4}\right) \\
\sin\left(\frac{3\pi}{4}\right) & \cos\left(\frac{3\pi}{4}\right)
\end{bmatrix}
\]
Thus, exploring other options is advised to confirm the correct matrix for this specific angle of rotation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9978ffa8-8e6a-4550-8363-e044b6a6e895%2Fed244b84-478c-429a-b4f9-bdad7021fa47%2Fpj2rx0q_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Rotation Matrix for \( \frac{3\pi}{4} \) Angle**
**Description:**
The task is to identify the matrix that performs a rotation by an angle of \( \frac{3\pi}{4} \).
**Matrix:**
\[
\begin{bmatrix}
-\frac{1}{2}\sqrt{2} & -\frac{1}{2}\sqrt{2} \\
\frac{1}{2} & -\frac{1}{2}\sqrt{2}
\end{bmatrix}
\]
**Explanation:**
This image shows a 2x2 matrix surrounded by a border indicating it was selected, but marked with an 'x', suggesting the selection is incorrect. For a rotation by \( \frac{3\pi}{4} \), the correct matrix is typically derived from:
\[
\begin{bmatrix}
\cos\left(\frac{3\pi}{4}\right) & -\sin\left(\frac{3\pi}{4}\right) \\
\sin\left(\frac{3\pi}{4}\right) & \cos\left(\frac{3\pi}{4}\right)
\end{bmatrix}
\]
Thus, exploring other options is advised to confirm the correct matrix for this specific angle of rotation.
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