Q37E Consider a plane V聽in R3聽with ... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q37E (page 217)

Consider a plane Vin $R^{3}$ with orthonormal basis ${\overset{鈫�}{u}}_{1}, {\overset{鈫�}{u}}_{2}$ . Let $\overset{鈫�}{x}$ be a vector in $R^{3}$ . Find a formula for the reflection $R (\overset{鈫�}{x})$ of $\overset{鈫�}{x}$ about the plane V.

Short Answer

Expert verified

The formula of the reflection is $R (\overset{鈫�}{x}) = 2 ({\overset{鈫�}{u}}_{1} 鈰� \overset{鈫�}{x}) {\overset{鈫�}{u}}_{1} + 2 ({\overset{鈫�}{u}}_{2} 鈰� \overset{鈫�}{x}) {\overset{鈫�}{u}}_{2} 鈭� \overset{鈫�}{x}$

Step by step solution

Consider the diagram below

Consider the below diagram for the given information.

Consider the diagram below

From the above diagram, it is clear that the distance of the point from the plane will be the same as the point after reflection. Thus,

$\begin{matrix} R (\overset{鈫�}{x}) = 2 p r o j_{V} \overset{鈫�}{x} + p r o j_{V} \overset{鈫�}{x} 鈭� \overset{鈫�}{x} \\ = 2 p r o j_{V} \overset{鈫�}{x} 鈭� \overset{鈫�}{x} \\ = 2 ({\overset{鈫�}{u}}_{1} 鈰� \overset{鈫�}{x}) {\overset{鈫�}{u}}_{1} + 2 ({\overset{鈫�}{u}}_{2} 鈰� \overset{鈫�}{x}) {\overset{鈫�}{u}}_{2} 鈭� \overset{鈫�}{x} \end{matrix}$

Hence, the value of the reflection $R (\overset{鈫�}{x})$ of $\overset{鈫�}{x}$ about the plane V is .

$R (\overset{鈫�}{x}) = 2 ({\overset{鈫�}{u}}_{1} 鈰� \overset{鈫�}{x}) {\overset{鈫�}{u}}_{1} + 2 ({\overset{鈫�}{u}}_{2} 鈰� \overset{鈫�}{x}) {\overset{鈫�}{u}}_{2} 鈭� \overset{鈫�}{x}$

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91影视

Consider a plane Vin $R^{3}$ with orthonormal basis ${\overset{鈫�}{u}}_{1}, {\overset{鈫�}{u}}_{2}$ . Let $\overset{鈫�}{x}$ be a vector in $R^{3}$ . Find a formula for the reflection $R (\overset{鈫�}{x})$ of $\overset{鈫�}{x}$ about the plane V.

Short Answer

Step by step solution

Consider the diagram below

Consider the diagram below

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91影视

Consider a plane Vin R3with orthonormal basis u鈫�1,u鈫�2. Let x鈫� be a vector inR3 . Find a formula for the reflection R(x鈫�) of x鈫�about the plane V.

Short Answer

Step by step solution

Consider the diagram below

Consider the diagram below

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Logic and Functions

Probability and Statistics

Geometry

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Consider a plane Vin $R^{3}$ with orthonormal basis ${\overset{鈫�}{u}}_{1}, {\overset{鈫�}{u}}_{2}$ . Let $\overset{鈫�}{x}$ be a vector in $R^{3}$ . Find a formula for the reflection $R (\overset{鈫�}{x})$ of $\overset{鈫�}{x}$ about the plane V.