I've been asked to prove that the set of pairs of unit vectors from $\mathbb R^3$, $(x, y)$ such that $x$ and $y$ are orthogonal is a manifold. The question isn't specific on which topology the set has, so I'm assuming the topology is either inherited from the metric on $\mathbb R^6$ or else is something equivalent.
My first instinct was to prove it directly. As a subspace of $\mathbb R^6$, Hausdorff and Second Countable are immediate that way, but I don't feel like I have a good enough grip on what neighborhoods look like in this space to exhibit a local homeomorphism at each point.
My other thought was showing the whole space is homeomorphic to a known manifold. I'm not sure I understand the space enough to do that correctly, but I've played with it, and it feels like it could be homeomorphic to $S^2 \times S^1$, because once I select the first vector, there is a circle's worth of second vectors to choose. But I'm not sure enough about the specifics on that to feel good about it.
Am I on the right track?