We consider the distributed source coding system of two correlated Gaussian Vector sources Y_l=^t(Y_l1, Y_l2),l=1,2 which are noisy observations of correlated Gaussian scalar source X₀. We assume that for each (l,k)∈{1,2}, Y_lk is an observation of the source X₀, having the form Y_lk=X₀+N_lk, where N_lk is a Gaussian random variable independent of X₀. We further assume that N_lk, (l,k)∈{1,2}² are independent. In this system two correlated Gaussian observations are separately compressed by two encoders and sent to the information processing center. We study the remote source coding problem where the decoder at the center attempts to reconstruct the remote source X₀. The determination problem of the rate distortion region for this communication system can be regarded as an extension of the Gaussian CEO problem to the case of vector observations. For each vector observation we can obtain an estimation on X₀ from this observation. Those estimations are sufficient statistics on X₀. Using those sufficient statistics, we determine the rate distortion region by showing that it coincides with the rate distortion region of the CEO problem where the scalar observations of X₀ are equal to the estimations computed from the vector observations. We further extend the result to the case of L terminal and general vector observations.