We show self-coupled and cross-coupled vector beam evolution equations in the low-amplitude regime for screening solitons, which can exhibit the analytical solutions of bright-bright and dark-dark vector solitons. Our analysis indicates that these self-coupled vector solitons are obtained irrespective of the intensities of the two optical beams, whereas these cross-coupled vector solitons can be established when the intensities of the two optical beams are equal. Relevant examples are provided where the photorefractive crystal is lithium niobate (LiNbO3). The stability properties of these vector solitons have been investigated numerically and it has been found that they are stable.