  Let v_1,...,v_{n-1} be n-1 independent vectors in R^n (or C^n). We study x,
the unit normal vector of the hyperplane spanned by the v_i. Our main finding
is that x resembles a random vector chosen uniformly from the unit sphere,
under some randomness assumption on the v_i.
  Our result has applications in random matrix theory. Consider an n by n
random matrix with iid entries. We first prove an exponential bound on the
upper tail for the least singular value, improving the earlier linear bound by
Rudelson and Vershynin. Next, we derive optimal delocalization for the
eigenvectors corresponding to eigenvalues of small modulus.

Probability