  In this paper, we study the random matrix model of Gaussian Unitary Ensemble
(GUE) with fixed-rank (aka spiked) external source. We will focus on the
critical regime of the Baik-Ben Arous-P\'ech\'e (BBP) phase transition and
establish the distribution of the eigenvectors associated with the leading
eigenvalues. The distribution is given in terms of a determinantal point
process with extended Airy kernel. Our result can be regarded as an eigenvector
counterpart of the BBP eigenvalue phase transition (arXiv:math/0403022). The
derivation of the distribution makes use of the recently re-discovered
eigenvector-eigenvalue identity, together with the determinantal point process
representation of the GUE minor process with external source.
