I do not have any idea about how to prove this:
Let $M = \{(x,y,z) \mid x^2 + y^2 - z^2 = 1\} \subset \mathbb{R}^3$. Show that $M$ is equal to the disjoint union of infinitely many straight lines.
This is a hyperboloid and based on its graph it looks like it is certainly the disjoint union of infinitely many straight lines. How can I prove this?
Write them down!
The plane $x=1$ meets $M$ in $y^2-z^2=0$, that is $y-z=0$ or $y+z=0$. The first of these lines is parametrically $(1,t,t)$. Rotating this about the $z$-axis gives $(\cos\theta+t\sin\theta,-\sin\theta+t\cos\theta,t)$. This is a parametrisation of the surface, and each fixed $\theta$ gives a line.