Problem 23. Consider the plane P in R³ defined by the equation x+2y+3z = 0 and the line L in R³ spanned [9] by the vector 3 Let S R3 R³ denote the reflection through the plane P: it takes a vector in R³ and : transforms it into its mirror image, the mirror being the plane P. Let T: R³ → R³ denote the 90° rotation around L of your choice (i.e. you can choose if the rotation is clockwise or counterclockwise). Both S and T are linear transformations (you don't have to prove that). Find the matrix A such that (ToS)(x) = Ax for all vectors R³. Here is some information that you might find useful: The vector 2 is perpendicular to the plane. -21] are perpendicular to the line. They are also perpendicular to each other. The vectors and -7 30 M2x2 T Maxz DB [-] B DB "K [T] B-B B-world

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter1: Vectors
Section1.3: Lines And Planes
Problem 18EQ
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do not use 

Rodrigues' rotation formula and reflection matrix formula. Instead, draw a diagram (example uploaded).

Problem 23. Consider the plane P in R³ defined by the equation x+2y+3z = 0 and the line L in R³ spanned
[9]
by the vector 3 Let S R3 R³ denote the reflection through the plane P: it takes a vector in R³ and
:
transforms it into its mirror image, the mirror being the plane P. Let T: R³ → R³ denote the 90° rotation
around L of your choice (i.e. you can choose if the rotation is clockwise or counterclockwise). Both S and T
are linear transformations (you don't have to prove that). Find the matrix A such that (ToS)(x) = Ax for all
vectors R³. Here is some information that you might find useful:
The vector 2 is perpendicular to the plane.
-21]
are perpendicular to the line. They are also perpendicular to each other.
The vectors
and
-7
30
Transcribed Image Text:Problem 23. Consider the plane P in R³ defined by the equation x+2y+3z = 0 and the line L in R³ spanned [9] by the vector 3 Let S R3 R³ denote the reflection through the plane P: it takes a vector in R³ and : transforms it into its mirror image, the mirror being the plane P. Let T: R³ → R³ denote the 90° rotation around L of your choice (i.e. you can choose if the rotation is clockwise or counterclockwise). Both S and T are linear transformations (you don't have to prove that). Find the matrix A such that (ToS)(x) = Ax for all vectors R³. Here is some information that you might find useful: The vector 2 is perpendicular to the plane. -21] are perpendicular to the line. They are also perpendicular to each other. The vectors and -7 30
M2x2
T
Maxz
DB
[-] B
DB
"K
[T] B-B
B-world
Transcribed Image Text:M2x2 T Maxz DB [-] B DB "K [T] B-B B-world
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