  We derive a system of nonpolynomial Schroedinger equations (NPSEs) for
one-dimensional wave functions of two components in a binary self-attractive
Bose-Einstein condensate loaded in a cigar-shaped trap. The system is obtained
by means of the variational approximation, starting from the coupled 3D
Gross-Pitaevskii equations and assuming, as usual, the factorization of 3D wave
functions. The system can be obtained in a tractable form under a natural
condition of symmetry between the two species. A family of vector
(two-component) soliton solutions is constructed. Collisions between orthogonal
solitons (ones belonging to the different components) are investigated by means
of simulations. The collisions are essentially inelastic. They result in strong
excitation of intrinsic vibrations in the solitons, and create a small
orthogonal component ("shadow") in each colliding soliton. The collision may
initiate collapse, which depends on the mass and velocities of the solitons.
