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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 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 (TS)(x) = Añ for all vectors R³. Here is some information that you might find useful: • The vector 2 is perpendicular to the plane. • The vectors 30 and -7 30 are perpendicular to the line. They are also perpendicular to each other.

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 (TS)(x) = Añ for all vectors R³. Here is some information that you might find useful: • The vector 2 is perpendicular to the plane. • The vectors 30 and -7 30 are perpendicular to the line. They are also perpendicular to each other.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.4: Plane Curves And Parametric Equations
Problem 33E
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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 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 (TS)(x) = Añ for all vectors R³. Here is some information that you might find useful: • The vector 2 is perpendicular to the plane. • The vectors 30 and -7 30 are perpendicular to the line. They are also perpendicular to each other.
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 (TS)(x) = Añ for all vectors R³. Here is some information that you might find useful: • The vector 2 is perpendicular to the plane. • The vectors 30 and -7 30 are perpendicular to the line. They are also perpendicular to each other.
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