We introduce the distributed estimation of a random vector signal in wireless sensor networks that follow coherent multiple access channel model. We adopt the linear minimum mean squared error fusion rule. The problem of interest is to design linear coding matrices for those sensors in the network so as to minimize mean squared error of the estimated vector signal under a total power constraint. We show that the problem can be formulated as a convex optimization problem and we obtain closed form expressions of the coding matrices. Numerical results are used to illustrate the performance of the proposed method.