Solution
= Solution
The points $(x,y)$ are ordered orthonormal two-frames, so $P$ is the <Stiefel manifold> $V_2(\mathbb R^{n+1})$. Changing an orthonormal basis of the same plane gives the displayed free right $O(2)$-action. Since $O(2)$ is compact, the action is proper, and the quotient is the <Grassmannian> of unoriented two-planes. The <Free proper Lie-group action theorem> makes
$$
P\longrightarrow P/O(2)
$$
a principal $O(2)$-bundle.