
Isometries of R3
I am reading Aigli Papantonopoulou's book Algebra: Pure and applied.
Exercise 18 of Chapter 14:Symmetries reads as follows:
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"Describe all the isometries of that map the point (1,1,1) to (1,0,0) and express them in the form as in Theorem 14.21, and in terms of S( ) = A + b as in Corollary 14.22
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I would really appreciate help with this problem.
Peter
Details from Papantonopoulou follow:
On pages 462  463 we find:
Possible isometries of or include:
(1) Translations: These are of the form = + b
(2) Rotations: These are of the form for some A SO(n). We write for a rotation by an angle (about some axis through the origin)
(3) Reflections: These are given by for some A O(n) with det A = 1, that is A not in SO(n). We write r for reflections given by the matrices
O(2)
O(3)
Since SO(n) is a subgroup of O(n) of index 2, every reflection is of the form r
Theorem 14.21
An isometry S of or can be uniquely expressed as
S = where i = 0 or 1
Corollary 14.22
An isometry S of or can be uniquely expressed as
S( ) = A + b where A O(n) and b

Re: Isometries of R3
the only part of this worth investigating is the translation part, because any element of O(3) preserves the origin. so ask yourself: if x>x+b takes (1,1,1) to (1,0,0), what must the 3vector b be?