Show that the transformation T: R2→R? that reflects points through the horizontal x,-axis and then reflects points through the line x, = x, is merely a rotation about the origin. What is the angle of rotation? Use this plotted point to construct T(e2). T(e2) = -1 (Type an integer or simplified fraction for each matrix element.) Use the transformed columns to construct A. 0 - 1 A = 1 (Type an integer or simplified fraction for each matrix element.) cos o - sin o Compare this matrix to the R2-R? rotation matrix, , to determine the angle of rotation, p. cos o sin o (Simplify your answer. Type your answer in radians. Use angle measures greater than or equal to 0 and less than 2n.)
Show that the transformation T: R2→R? that reflects points through the horizontal x,-axis and then reflects points through the line x, = x, is merely a rotation about the origin. What is the angle of rotation? Use this plotted point to construct T(e2). T(e2) = -1 (Type an integer or simplified fraction for each matrix element.) Use the transformed columns to construct A. 0 - 1 A = 1 (Type an integer or simplified fraction for each matrix element.) cos o - sin o Compare this matrix to the R2-R? rotation matrix, , to determine the angle of rotation, p. cos o sin o (Simplify your answer. Type your answer in radians. Use angle measures greater than or equal to 0 and less than 2n.)
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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![Show that the transformation \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) that reflects points through the horizontal \( x_1 \)-axis and then reflects points through the line \( x_2 = x_1 \) is merely a rotation about the origin. What is the angle of rotation?
**Use this plotted point to construct \( T(e_2) \):**
\[ T(e_2) = \begin{bmatrix} -1 \\ 0 \end{bmatrix} \]
(Type an integer or simplified fraction for each matrix element.)
**Use the transformed columns to construct \( A \).**
\[ A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \]
(Type an integer or simplified fraction for each matrix element.)
**Compare this matrix to the \( \mathbb{R}^2 \to \mathbb{R}^2 \) rotation matrix,**
\[ \begin{bmatrix} \cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{bmatrix} \]
to determine the angle of rotation, \( \phi \).
\[ \phi = \boxed{} \]
(Simplify your answer. Type your answer in radians. Use angle measures greater than or equal to 0 and less than \( 2\pi \).)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fff96cdc2-7807-4421-a794-d30f1f1244f8%2F09cbefe6-b9f3-4860-9716-771e0d355218%2Fd344a8h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Show that the transformation \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) that reflects points through the horizontal \( x_1 \)-axis and then reflects points through the line \( x_2 = x_1 \) is merely a rotation about the origin. What is the angle of rotation?
**Use this plotted point to construct \( T(e_2) \):**
\[ T(e_2) = \begin{bmatrix} -1 \\ 0 \end{bmatrix} \]
(Type an integer or simplified fraction for each matrix element.)
**Use the transformed columns to construct \( A \).**
\[ A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \]
(Type an integer or simplified fraction for each matrix element.)
**Compare this matrix to the \( \mathbb{R}^2 \to \mathbb{R}^2 \) rotation matrix,**
\[ \begin{bmatrix} \cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{bmatrix} \]
to determine the angle of rotation, \( \phi \).
\[ \phi = \boxed{} \]
(Simplify your answer. Type your answer in radians. Use angle measures greater than or equal to 0 and less than \( 2\pi \).)
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