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Problem 23. Consider the plane P in R³ defined by the equation x+2y+32= = O and the line L in R³ 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. -21 The vectors 3 and 0 -7 30 are perpendicular to the line. They are also perpendicular to each other. Finally, here is a sugestion: don't hesitate to use a 3D graphing calculator (like Desmos) to help you picture the plane and the line.

Problem 23. Consider the plane P in R³ defined by the equation x+2y+32= = O and the line L in R³ 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. -21 The vectors 3 and 0 -7 30 are perpendicular to the line. They are also perpendicular to each other. Finally, here is a sugestion: don't hesitate to use a 3D graphing calculator (like Desmos) to help you picture the plane and the line.

Linear Algebra: A Modern Introduction
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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Problem 23. Consider the plane P in R³ defined by the equation x+2y+32= = O and the line L in R³ 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. -21 The vectors 3 and 0 -7 30 are perpendicular to the line. They are also perpendicular to each other. Finally, here is a sugestion: don't hesitate to use a 3D graphing calculator (like Desmos) to help you picture the plane and the line.
Transcribed Image Text:Problem 23. Consider the plane P in R³ defined by the equation x+2y+32= = O and the line L in R³ 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. -21 The vectors 3 and 0 -7 30 are perpendicular to the line. They are also perpendicular to each other. Finally, here is a sugestion: don't hesitate to use a 3D graphing calculator (like Desmos) to help you picture the plane and the line.
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