What is the exact definition of a reflection through the plane $a.r=0$ for a given vector a and $r=(x,y,z)$. Of course I know what it is but I don't know what's part of its definition and what's part of its properties anymore.
My aim is to prove that this type of reflection is linear and that $R(u).R(u)=u.u$, can you help me? It seems so obvious that I can't actually prove it...
Thank you
On
Given a dot product and a nonzero vector $a,$ the reflection $\tau_a$ is given by $$ \tau_a (x) = x - \left( \frac{2 a \cdot x}{a \cdot a}\right) a $$ Straightforward to calculate $\tau_a (x) \cdot \tau_a(x).$
Note that a reflection, along with being an isometry, is also self adjoint, meaning its matrix will be symmetric as well as orthogonal; we write this as $$ \tau_a (x) \cdot y = x \cdot \tau_a(y) $$If this seems peculiar (it does to me), consider that, given the reflection matrix $R,$ there is some orthogonal matrix $P,$ meaning $P^T P = I,$ such that $P^T R P$ is a diagonal matrix with $n-1$ diagonal entries equal to $1$ and the final diagonal entry equal to $-1.$
It is the unique isometry of $R^3$ whose fixed point set is exactly the plane $a\cdot r=0$.