Let $V$ be an $n$-dimensional real inner product space. Suppose $u \in V$ has norm 1, and define the reflection in the direction of $u$ as $$r_u(v) = v - 2 \langle v, u \rangle u = v - 2proj_{span(u)}(v)$$ for all $v \in V$.
a) What is the trace and determinant of $r_u$?
b) Show that $r_u$ is an isometry.
c) Show that if $v, w \in V$, then there exists a unique $r_u \in \mathcal{L}(V)$ for which $r_u(v)=w$.
For (a), I tried solving for the determinant and trace on a 3- and 4-dim vector space $V$, but I wasn't sure how to generalize to $n$ dimensions. As for (b) and (c), I'm unsure on where to begin. Any help and guidance would be lovely, thanks!
EDIT: Fixed question (c).
a) The trace and determinant can be computed if you know the eigenvalues of $r_u$. (Hint: the eigenvalues are $-1$ and $1$, but you need to figure out how many of each there are.)
b) Have you written down the definition of isometry and tried to verify that $r_u$ satisfies it?
c) Draw a picture (say, in 2-dimensions) of some arbitrary $v$ and $w$, and think about what kind of reflection sends $v$ to $w$. Can you draw what $u$ should be for that reflection? (Hint: the line in direction $u$ should perpendicularly bisect the segment between $v$ and $w$.)