# hw6sol - Homework 6 Solutions 1 Show that if is a...

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Homework 6 Solutions 1. Show that if τ is a translation, then there exists an odd isometry β so that τ = β 2 . Solution: Let τ = τ ~v . Now, choose a line which is parallel to ~v . We claim that γ 2 ‘,~v/ 2 = τ . To verify this, let P be an arbitrary point in R 2 . Let Q be the point on which is closest to P and let ~u be the vector from Q to P . Note that by construction ~u is perpendicular to ‘,~v . We now ﬁnd γ ‘,~v/ 2 γ ‘,~v/ 2 ( P ) = τ ~v/ 2 σ τ ~v/ 2 σ ( P ) = τ ~v/ 2 σ τ ~v/ 2 ( P - 2 ~u ) = τ ~v/ 2 σ ( P - 2 ~u + ~v/ 2) = τ ~v/ 2 ( P + ~v/ 2) = P + ~v It follows that γ 2 ‘,~v/ 2 = τ ~v as desired. 2. Consider a glide reﬂection γ ‘,~v . Let P be a point on . Prove that there exists a point Q not on so that P is the midpoint of Q and γ ‘,~v ( Q ). Solution: Choose a vector ~u which is perpendicular to ‘,~v . Now, let Q = P - ~v/ 2 + ~u . Now we have Q 0 = γ ‘,~v ( Q ) = τ ~v σ ( Q ) = τ ~v σ ( P -

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## This note was uploaded on 01/30/2012 for the course MATH 310 303 taught by Professor Rwk during the Spring '11 term at Simon Fraser.

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hw6sol - Homework 6 Solutions 1 Show that if is a...

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