466-HW-5 - ρ D Φ such that τ A,B = ρ D Φ ρ C,θ 7 State if the following are True or False If true give a short proof Otherwise give a

Math 466 - Homework Set 5 (8.12.2011) 1. Let ‘ be the line y = 0 . Find a formula for the reﬂection σ ‘ . 2. Let ‘ be the line x = 0 . Find a formula for the reﬂection σ k . 3. Let k be the line y = mx where m = tan θ . Find a formula for the reﬂection σ k as follows : (a) Rotate the line n clockwise about the origin through θ radians. If ρ O, - θ ( x,y ) = ( x 000 ,y 000 ) ﬁnd a formula for this rotation. (b) Use the formula you found in question 1 to ﬁnd the reﬂection of the rotated line. If σ ‘ ( x 000 ,y 000 ) = ( x 00 ,y 00 ) write the formula for this reﬂection. (c) Use the inverse of the rotation you found in (a) to go back. If ρ O,θ ( x 00 ,y 00 ) = ( x 0 ,y 0 ) ﬁnd a formula for this rotation. (d) Explain why ρ O,θ σ ‘ ρ O, - θ = σ k . 4. Let ‘ be the line with an equation y = mx + b ( m 6 = 0; b 6 = 0) . Note that ‘ intersects the x - axis at the point ( - b/m, 0) . Use the formula you found in question 3 to ﬁnd a formula for the reﬂection in ‘ . 5. With the notation as in Figure 1 show that ρ C, 2Ψ ρ B, 2Φ ρ A, 2 θ is a non- identity translation although ρ A, 2 θ ρ B, 2Φ ρ C, 2Ψ = ι . 6. Given τ A,B and nonidentity rotation ρ C,θ is it possible to ﬁnd a rotation

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Unformatted text preview: ρ D, Φ such that τ A,B = ρ D, Φ ρ C,θ ? 7. State if the following are True or False . If true give a short proof. Otherwise give a counterexample (whenever it is possible to do so). (a) σ P σ ‘ σ P σ ‘ σ P σ ‘ σ P is a reﬂection in a line parallel to line ‘ . (b) ρ B, Φ ρ A,-Φ is the translation that takes A to ρ B, Φ ( A ) . (c) Every isometry is a product of two involutions. (d) If n > 2 then D n is generated by two reﬂections. 1 (e) A group of isometries of order 143 must be a cyclic group C 143 . 8. What are the equations for ρ O,π/ 3 and ρ (1 ,-2) ,π/ 3 ? 9. What is the test for the symmetry with respect to the line ‘ with an equation y = 2 x-1 ? 10. Let ‘ be the line with an equation of the form y = mx + b and P = ( a,ma + k ) be a point . If f = τ O,P σ ‘ ﬁnd equations for the glide reﬂection f by using question 4. 2...
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