$P,Q\in \ell_1$ which intersects the unit sphere at $P,Q$. Show that the planes tangent to the unit sphere at $P,Q$ intersect along a line $\ell$ $$ \ell_1: \begin{cases} x=-1+t\\ y=0+0t\\ z=2-t \end{cases} $$
I have no idea what to do in this exercise. Help is much appreciated.
Where did you get stuck? It is hard to believe that you had "no idea", as the problem can be broken down to smaller questions in a straightforward way. Compute $P, Q$ first. This is easy: first of all you have the system of equations, and you also have the equation of the unit sphere: $x^2+y^2+z^2=1$.
Can you solve this system of equtions?
Once you have $P$ and $Q$, compute the tangent planes (again, straightforward), and then compute the intersection of them as a solution to a system of linear equations.
There are even simpler methods to directly answer your question, but this one is absolutely straightforward, and it requires no idea at all.