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Homework 7
1. Let
ρ
P,θ
be a rotation and let
β
be any isometry. Prove that
βρ
P,θ
β

1
is a rotation.
Solution:
Let
Q
=
β
(
P
) so
P
=
β

1
(
Q
). Then we ﬁnd
βρ
P,θ
β

1
(
Q
) =
βρ
P,θ
(
P
) =
β
(
P
) =
Q
.
So
Q
is a ﬁxed point of
βρ
P,θ
β

1
. Since
ρ
P,θ
is even,
βρ
P,θ
β

1
(
Q
) is also even. Now,
βρ
P,θ
β

1
(
Q
) is an even isometry which ﬁxes the point
Q
, so it must be a rotation about
Q
(which might be the identity).
2. Let
τ
~v
be a translation and let
β
be any isometry. Prove that
βτ
~v
β

1
is a translation
(Hint: use 1.)
Solution:
The map
βτ
~v
β

1
is an even isometry so it must be a rotation or translation.
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 Spring '11
 RWK

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