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**Unformatted text preview: **ℓ and m such that σ ℓ σ m is the rotation about (1 , 1) with angle π 3 . ( ii ) Describe the isometry σ ℓ σ m when ℓ is the line x + 2 y = 0 and m is the line 2 x-y = 0. 6. ( i ) ²ind equations for a pair of lines ℓ and m such that σ ℓ σ m is the translation τ u where u = b 1 √ 3 B . ( ii ) Describe the isometry σ ℓ σ m when ℓ is the line x + y = 1 and m is the line x + y = 3. 7. By expressing each of α and β suitably as a composite of two re±ections, show the following. ( i ) If α and β are rotations then either αβ is a rotation or αβ is a translation. ( ii ) If α is a rotation and β is a translation then αβ is a rotation. 8. Show that the set of all translations and rotations is a group....

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