# Thread: Isometries of R3

1. ## 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 $\mathbb{R}^3$ that map the point (1,1,1) to (1,0,0) and express them in the form $t_b$ $\circ$ $\rho_{\theta}$ $\circ$ $r^i$ as in Theorem 14.21, and in terms of S( $\nu$) = A $\nu$ + 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 $\mathbb{R}^2$ or $\mathbb{R}^3$ include:

(1) Translations: These are of the form $t_b(\nu)$ = $\nu$ + b

(2) Rotations: These are of the form $T_A$ for some A $\in$SO(n). We write $\rho_{\theta}$ for a rotation by an angle $\theta$ (about some axis through the origin)

(3) Reflections: These are given by $T_A$ for some A $\in$O(n) with det A = -1, that is A not in SO(n). We write r for reflections given by the matrices

$\left(\begin{array}{cc}1&0\\0&-1\end{array}\right)$ $\in$ O(2)

$\left(\begin{array}{ccc}1&0&0\\0&1&0\\0&0&-1\end{array}\right)$ $\in$ O(3)

Since SO(n) is a subgroup of O(n) of index 2, every reflection is of the form $\rho_{\theta}$ $\circ$r

Theorem 14.21

An isometry S of $\mathbb{R}^2$ or $\mathbb{R}^3$can be uniquely expressed as

S = $t_b$ $\circ$ $\rho_{\theta}$ $\circ$ $r^i$ where i = 0 or 1

Corollary 14.22

An isometry S of $\mathbb{R}^2$ or $\mathbb{R}^3$can be uniquely expressed as

S( $\nu$) = A $\nu$ + b where A $\in$ O(n) and b $\in$ $\mathbb{R}^3$

2. ## 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 3-vector b be?