Where are the headers of the C++ standard library

Running g++ -v -v -v outputs lots of things, including all the include directories searched. vector is in one of those.

On my Debian Stable system vector is here:

/usr/include/c++/4.4/vector

GCC typically has the standard C++ headers installed in /usr/include/c++/<version>/. You can run gcc -v to find out which version you have installed.

At least in my version, there is no vector.h; the public header is just vector (with no extension), and most of the implementation is in bits/stl_vector.h.

That's the case on my Ubuntu distribution; your distribution may differ.