In an orthonormal coordinate system $K=O \overrightarrow{e_1}\overrightarrow{e_2} \overrightarrow{e_3}$ I have to find an analytic representation of an axis symmetry with an axis g represented by two points $A(0,1,0)$ and $B(3,1,-4)$.
What I have so far:
I know that symmetry can be represented as a rotation with angle $\phi=180^o$ and I know the matrix of that. I guess I also have to represent it in a different coordinate system $O^*$ where the basis vector $\overrightarrow{O^*e_3^*}\equiv g_{AB}$.
So, $|\overrightarrow{e_3^*}|=1$, in that case $\overrightarrow{e_3^*}(\frac{3}{5}, 0, -\frac{4}{5}),\, \overrightarrow{e_2^*}(0,1,0),\, \overrightarrow{e_1^*}(-\frac{4}{5}, 0, -\frac{3}{5})$. From those three vrectors I form the matrix for changing of basis $T$, and it's transponse $T^{-1}=T^t$, I use the matrix for rotation with angle $\phi=180^o$ denoted $A^*$, and I represent the whole transformation in our original coordinate system as a matrix $A=T \circ A^*\circ T^{-1}$.
Is this correct? I have the feeling that the center of the coordinate system is changed and I don't know how to express that.