Do truly absolute directions exist? If not, what directions exist independent of coordinate systems? I feel that, until I understand the answers to these questions, I will not fully understand how vectors exist independent of coordinate systems.

Just a big ol' slice down the middle. Would it drift apart over time, or eventually fuse back together?

If we add a constant term $k$ to the potential function in time-independent Schrödinger's equation, $V(x) \rightarrow V(x)+k$, then how does it affect the solution, and what is its significance? Especially, when it has no significance in classical mechanics.

According to my textbook: Moment of a force = force $\times$ perpendicular distance of the line of action of the force from the fulcrum. I don't understand why it would be that way (especially what mathematics goes behind it). Why is distance taken to be perpendicular to the direction of force ...