Find the matrix that produces the given rotation. 45° about the z-axis
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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Question
Find the matrix that produces the given rotation.
![**Topic: Rotation Matrix in 3D Space**
**Objective:** Find the matrix that produces the given rotation: 45° about the z-axis.
**Explanation:**
This exercise involves creating a rotation matrix for a specific rotation in three-dimensional space. The task is to find the matrix capable of rotating a vector 45 degrees around the z-axis.
**Rotation Matrix Formula:**
A rotation about the z-axis by an angle θ is represented by the matrix:
\[
\begin{bmatrix}
\cos(\theta) & -\sin(\theta) & 0 \\
\sin(\theta) & \cos(\theta) & 0 \\
0 & 0 & 1
\end{bmatrix}
\]
For a 45° (π/4 radians) rotation, the cosine and sine values are:
- \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\)
- \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\)
Therefore, the rotation matrix becomes:
\[
\begin{bmatrix}
\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 0 \\
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\
0 & 0 & 1
\end{bmatrix}
\]
**Conclusion:**
Using these values, fill in the matrix to complete the rotation transformation for a 45° rotation about the z-axis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7d6fc442-7ad3-4518-9add-f310e8d32dde%2F06ae69b3-10ab-4962-b6a4-203d9cd7bfbc%2F5sa64as_processed.png&w=3840&q=75)
Transcribed Image Text:**Topic: Rotation Matrix in 3D Space**
**Objective:** Find the matrix that produces the given rotation: 45° about the z-axis.
**Explanation:**
This exercise involves creating a rotation matrix for a specific rotation in three-dimensional space. The task is to find the matrix capable of rotating a vector 45 degrees around the z-axis.
**Rotation Matrix Formula:**
A rotation about the z-axis by an angle θ is represented by the matrix:
\[
\begin{bmatrix}
\cos(\theta) & -\sin(\theta) & 0 \\
\sin(\theta) & \cos(\theta) & 0 \\
0 & 0 & 1
\end{bmatrix}
\]
For a 45° (π/4 radians) rotation, the cosine and sine values are:
- \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\)
- \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\)
Therefore, the rotation matrix becomes:
\[
\begin{bmatrix}
\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 0 \\
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\
0 & 0 & 1
\end{bmatrix}
\]
**Conclusion:**
Using these values, fill in the matrix to complete the rotation transformation for a 45° rotation about the z-axis.
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