s12_mthsc481_hw02

s12_mthsc481_hw02 - Homework 2 | Due January 31(Tuesday 1...

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Unformatted text preview: Homework 2 | Due January 31 (Tuesday) 1 Read : Stahl, Chapters 2.2, 2.3, 2.4, 2.5, 3.1. 1. In this problem, use only the analytic definition of Euclidean geometry. Let v be a unit vector in R 2 , and define r v ( x ) = x- 2( x · v ) v . (a) Prove (analytically) that r v is a Euclidean isometry. (b) Prove that r v is the reflection in a line m through the origin, and find an equation for that line (both in y = mx + b , and in parametric form). (c) For an arbitrary α ∈ R , describe the isometry f := τ α v ◦ r v ◦ τ- 1 α v geometrically, and prove your description. This is the unique map that makes the following diagram commute: E 2 r v / τ α v E 2 τ α v E 2 f / _ _ _ _ _ _ E 2 What do r v and τ α v ◦ r v ◦ τ- 1 α v have in common? In the following three problems, you will prove a series of results stated in Chapter 2 of Stahl. For each problem, you are free to use results in Stahl that appear before it, provided you clearly state and cite them. The notation ρ m denotes the reflection of...
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s12_mthsc481_hw02 - Homework 2 | Due January 31(Tuesday 1...

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