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
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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.
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: x, y, z.

If the point: x, y, z is reflected about the x-z plane, it becomes x, -y, z.

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