Show that an orthogonal transformation from $\mathbb{R^3}$ to $\mathbb{R^3}$ can be written as the composition of reflections If $T$ is an orthogonal transformation from $\mathbb{R^3}$ to $\mathbb{R^3}$, then $T$ may be written as the composition of three or fewer reflections about planes in $\mathbb{R^3}$.
Suppose $\{x_1,x_2,e_3\}$ is an orthonormal set in $\mathbb{R^3}$. Show that there is a plane $V$ such that $ref_V(x_2)=e_2$ and $ref_V(e_3)=e_3$.
Suppose $\{u_1,u_2,u_3\}$ is an orthonormal set in $\mathbb{R^3}$. Show that there is a plane $V$ such that $ref_V(u_3)=e_3$.
For the first one, I know the plane has to contain $e_3$, so we can say that the plane is spanned by $e_3$ and something else so that the two reflections hold, but what can this something else be?
For the second one, I don't see why $ref_V(u_3)=e_3$ would imply that the set is orthonormal (or is this not what the question is saying?).
 A: For the first question: Note that both $x_2$ and $e_2$ are perpendicular to $e_3$, so we can
start by considering the plane $\Bbb R^2 \DeclareMathOperator{\ref}{ref}$.  
Can you find a vector $v = (v_1,v_2)\in \Bbb R^2$ such that $\ref_v(e_2) = x_2$?  Consider a $v$ whose angle from the $x$-axis is exactly half-way between that of $e^2$ and $x_2$.  How could we construct such a vector (perhaps using rotation matrices)?
Now, let $V$ be the plane spanned by $(v_1,v_2,0)$ and $e_3$.  Verify that $\ref_V(e_2) = x_2$.

Now, for the second one: we are given that $u_1,u_2,u_3$ are orthonormal, so that is not what the question is saying.
If $u_3$ and $e_3$ point in the same direction, then set $V$ to be any plane containing $e_3$, since $e_3 = u_3$.  If they point in opposite directions (so that $u_3 = -e_3$),
set $V$ to be the plane spanned by $e_1$ and $e_2$ (the $xy$-plane).  
Otherwise, $u_3$ and $e_3$ span a plane.  Consider a rotation of this plane that takes $u_3$ to $e_3$.
Let $v_1$ be a vector perpendicular to this plane.
Let $v_2$ be the vector that comes from rotating $u_3$ half-way towards $e_3$.  Take $V$ to be the plane spanned by $v_1$ and $v_2$.
Verify in this case that $\ref_V(u_3) = e_3$.
