How to count all the orthogonal transformations?

[skrat]

Homework Statement

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!

Homework Equations

##A_{S}=PA_{0}B^{-1}##

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.

 

[tiny-tim]

hi skrat!

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

see the start of http://en.wikipedia.org/wiki/Orthogonal_transformation

 

[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?