Any two reflections of $S^n$ across different n-dimensional hyperplanes are homotopic through reflections. My question is about the Ex.2.2.5 of Hatcher's book.


  
*Show that any two reflections of $S^n$
  across different n-dimensional hyperplanes
  are homotopic, in fact homotopic through reflections. [The linear algebra formula for
  a reflection in terms of inner products may be helpful.]
  

I know the reflection formula. Let $a,b$ be two nonparallel unit normal vector of a hyperplane in $\mathbb R^{n+1}$. We want to prove that 
$f_a(x)=x-2<x,a>a$ and $f_b(y)=x-2<x,b>b$ are homotopic.(Note that $f_a(x)+f_b(x)$ may be zero, so the usual formula $tf_a(x)+(1-t)f_b(x)/\|tf_a(x)+(1-t)f_b(x)\|$ won't work here.)
I think we need some homology theory in here because this problem is taken from the chapter in homology theory. But I don't know where I am gonna use it.

I'm not sure you do need any homology theory for this! (Though the result is certainly relevant to homology since it confirms that any two reflections have the same degree as maps from $S^n$ to $S^n$.)
Let $\gamma: [0,1] \to S^n$ be any continuous path on the unit sphere such that $\gamma(0) = a$ and $\gamma(1) = b$. For any unit vector $u \in S^n$, let $f_u : S^n \to S^n$ denote the reflection in the plane perpendicular to $u$.
Then surely $F: [0,1] \times S^n \to S^n$ defined by
$F(t,x) = f_{\gamma(t)}(x)$ is a homotopy from $f_a$ to $f_b$?
(For example, this $F$ is continuous, by your linear algebra formula?)
