Computational complexity of generating a random vector
You can't create a uniformly random number in $[0,1]$ in finite time, because that would require infinite precision.
Probably that's not what you want.  Probably you want to generate a random float in the range $[0,1]$.  Then we have to ask what assumptions you are willing to make about your pseudorandom number generator. For many of them it is probably reasonable to treat generating a random float in that range as taking $O(1)$ time.  If so, you can create a $N$-dimensional vector with $N$ calls to that pseudorandom generator, i.e., in $O(N)$ time.
See How to come up with the runtime of algorithms?, How does one know which notation of time complexity analysis to use?, Is there a system behind the magic of algorithm analysis? and a good textbook for an overview of asymptotic running time analysis.