Show that the matrix $P=I-2hh^T$ is orthogonal and find its first column.

Question

Let $x=(x_1,...,x_n)^T$ a column vector in $\mathbb{R}^n$ so that $x_1\neq -1.$ Let $h$ a unitary vector in the direction of $x-e_1$ where $e_1$ is the vector in $\mathbb{R}^n$, $e_1=(1,0,...,0)$. Show that the matrix $P=I-2hh^T$ is orthogonal and find its the first column.

I showed that $P$ but I not can obtain its first column.

Thanks for your help!