# How to count all the orthogonal transformations?

1. Jun 1, 2013

### skrat

1. The problem statement, all variables and given/known data
Find an orthogonal transformation $\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ that map plane $x+y+z=0$ into $x-y-2z=0$ and vector $v_{1}=(1,-1,0)$ into $(1,1,0)$. Count all of them!

2. Relevant equations
$A_{S}=PA_{0}B^{-1}$

3. The attempt at a solution
So basis $B=\begin{bmatrix} 1 & 1 & 1\\ 1& -1 & 1\\ 1& 0& 2 \end{bmatrix}$ and $P=\begin{bmatrix} 1 & 1 & 1\\ -1& 1 & -1\\ -2& 0& 1 \end{bmatrix}$ where the last vector in both basis is a vector product of $n_{1} \times v_{1}$

$A_{0}=diag(1,1,1)$ and $A_{S}=PA_{0}B^{-1}$

Now, how do I count them? What do they represent - meaning, how do they differ from this one? please help.

2. Jun 1, 2013

3. Jun 1, 2013

### skrat

So whatever I do, $det(A_{0})=1$ has to stay the same, where $A_{0}=diag(1,1,1)$. Meanin I could also use $A_{0}=diag(-1,-1,1)$ or $A_{0}=diag(-1,1,-1)$ or $A_{0}=diag(1,-1,-1)$, right?

To sum up, there are 4 orthogonal transformations that do that?