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Problem 23. Consider the plane P in R³ defined by the equation x+2y+3z = 0 and the line L in R³ 9 . spanned 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 (TS)(x) = Ax for all vectors R³. Here is some information that you might find useful: The vector 2 is perpendicular to the plane. The vectors 130 and -21 -7 are perpendicular to the line. They are also perpendicular to each other. 30

Problem 23. Consider the plane P in R³ defined by the equation x+2y+3z = 0 and the line L in R³ 9 . spanned 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 (TS)(x) = Ax for all vectors R³. Here is some information that you might find useful: The vector 2 is perpendicular to the plane. The vectors 130 and -21 -7 are perpendicular to the line. They are also perpendicular to each other. 30

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Linear Transformations
Section6.5: Applications Of Linear Transformations
Problem 4E
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Problem 23. Consider the plane P in R³ defined by the equation x+2y+3z = 0 and the line L in R³ 9 . spanned 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 (TS)(x) = Ax for all vectors R³. Here is some information that you might find useful: The vector 2 is perpendicular to the plane. The vectors 130 and -21 -7 are perpendicular to the line. They are also perpendicular to each other. 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 9 . spanned 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 (TS)(x) = Ax for all vectors R³. Here is some information that you might find useful: The vector 2 is perpendicular to the plane. The vectors 130 and -21 -7 are perpendicular to the line. They are also perpendicular to each other. 30
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