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# t02sol - The University of Sydney Math3061 Geometry and...

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The University of Sydney Math3061 Geometry and Topology Web page: www.maths.usyd.edu.au/u/UG/SM/MATH3061/ 2009 Tutorial 2 1. Consider the set S consisting of the four transformations α , β , γ , ι given by α ( x,y ) = ( x, y ) , β ( x,y ) = ( x,y ) , γ ( x,y ) = ( x, y ) , ι ( x,y ) = ( x,y ) . ( i ) Describe α , β , γ , ι geometrically. ( ii ) Show that S is a group. Solution. ( i ) α is reflection in the x axis, β is reflection in the y axis, γ is the halfturn about O = (0 , 0), and ι is the identity transformation. ( ii ) α 1 = α,β 1 = β,γ 1 = γ and ι 1 = ι. So φ 1 S whenever φ S. Also, we have the following “multiplication table”. α β γ ι α ι γ β α β γ ι α β γ β α ι γ ι α β γ ι (To find the entry in the ’ α ’ row and ’ γ ’ column of the table, for instance, we calculate αγ ( x,y ) = α ( x, y ) = ( x,y ) = β ( x,y ) . Thus αγ = β , and this is placed in the table in the α row and γ column.) From this we see that φψ S whenever φ,ψ S . 2. Suppose that and m are lines, and that n = σ m ( ). Show that and n are parallel if and only if either is parallel to m or is perpendicular to m . Solution. Let X,Y be any two distinct points on . Choose a point Q on m and a unit vector u perpendicular to m . Then −−−−−−→ m ( X ) = 2( −−→ XQ. u ) u and −−−−−→ m ( Y ) = 2( −→ YQ. u ) u

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2 Therefore −−−−−−−−−→ σ m ( X ) σ m ( Y ) = −−−−−−→ σ m ( X ) X + −−→ XY + −−−−−→ m ( Y ) = 2( −−→ XQ. u ) u + −−→ XY + 2( −→ YQ. u ) u = 2( −−→ YX. u ) u + −−→ XY. So n = σ m ( X ) σ m ( Y ) is parallel to = XY ⇐⇒ either −−→ YX. u = 0 or u is a multiple of −−→ XY ⇐⇒ either is parallel to m or m .
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t02sol - The University of Sydney Math3061 Geometry and...

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