Find the matrix of the mirror reflection about the x-z plane in R³.
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 7**: Find the matrix of the mirror reflection about the \( x - z \) plane in \( \mathbb{R}^3 \).
**Explanation**:
To find the matrix of the mirror reflection about the \( x - z \) plane, consider that in \( \mathbb{R}^3 \) (three-dimensional space), a reflection over the \( x - z \) plane will reflect the \( y \)-coordinate. The coordinates \( x \) and \( z \) remain unchanged, while the sign of the \( y \)-coordinate is inverted.
Therefore, the transformation matrix for this reflection is:
\[
\begin{bmatrix}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 1
\end{bmatrix}
\]
This matrix, when multiplied by a vector \(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\), will result in a vector \(\begin{bmatrix} x \\ -y \\ z \end{bmatrix}\), which reflects the original vector across the \( x - z \) plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8cf9958e-dd38-41eb-b691-55a1e2252bf8%2F6f61c8f7-b7d0-495a-88ca-5b75377952f5%2F69tj3s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question 7**: Find the matrix of the mirror reflection about the \( x - z \) plane in \( \mathbb{R}^3 \).
**Explanation**:
To find the matrix of the mirror reflection about the \( x - z \) plane, consider that in \( \mathbb{R}^3 \) (three-dimensional space), a reflection over the \( x - z \) plane will reflect the \( y \)-coordinate. The coordinates \( x \) and \( z \) remain unchanged, while the sign of the \( y \)-coordinate is inverted.
Therefore, the transformation matrix for this reflection is:
\[
\begin{bmatrix}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 1
\end{bmatrix}
\]
This matrix, when multiplied by a vector \(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\), will result in a vector \(\begin{bmatrix} x \\ -y \\ z \end{bmatrix}\), which reflects the original vector across the \( x - z \) plane.
Expert Solution
Step 1
Any point on the three dimension can be represented using the point: .
If the point: is reflected about the x-z plane, it becomes .
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