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Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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.](https://content.bartleby.com/qna-images/question/7d6fc442-7ad3-4518-9add-f310e8d32dde/06ae69b3-10ab-4962-b6a4-203d9cd7bfbc/5sa64as_thumbnail.png)
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.
Expert Solution
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- e) Write the 3D Rotation Matrix that first rotates around the z-axis by 0₂ = 45° followed by a rotation on the y-axis by y = 45° followed by a rotation around the x-axis by 0x = 180° Rows: 1 Columns: 1 Carrow_forward2. Create a matrix expression that represents a 90°clockwise rotation of v reduced by Calculate the final vector. Be sure to show and explain all work. V=arrow_forwardLet A denote the standard matrix of rotation of R through 5° about the origin. Which one the following is the standard matrix of rotation of R? through -15° about the origin? O A inverse O A to the power of -3 O-3A O -15A A to the power of -15 A О ЗА O-A O A to the power of 3arrow_forward
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