The magnetization vector M was measured in the reversible region of the mixed state of a high quality Tl2Ba2Ca2Cu3O10 single crystal as a function of temperature and for different magnetic field amplitudes and orientations. These measurements allowed to study the M components perpendicular and parallel to the CuO2 layers (M ⊥ and M , respectively) under arbitrary values for the corresponding components of the applied magnetic field, H ⊥ and H . For temperatures close to Tc (in the critical fluctuation region) we observed M ⊥ (H ⊥ , H ) ≈ M ⊥ (H ⊥ , 0) and M (H ⊥ , H ) ≈ 0, as expected for an extremely anisotropic material. However, deviations from this behavior are observed at lower temperatures in the London region. In particular, the M ⊥ amplitude under a constant H ⊥ decreases on increasing H . In turn, in spite of the experimental uncertainties affecting M (mainly associated with the rotating sample holder), its amplitude under a constant H becomes observable on increasing H ⊥ . Both effects lead to an angular slippage of the magnetization vector M from the c crystallographic axis when the applied magnetic field is tilted from that axis. The M ⊥ behavior is phenomenologically explained in the framework of Lawrence-Doniach approaches for single layered superconductors by just assuming a slight angular dependence of the so-called vortex structure constant. However, the observed M is orders of magnitude larger than expected, a fact which is related to the multilayered nature of the compound studied. Our present results also directly affect the interpretation of recent measurements of the magnetic torque in other extremely anisotropic high-Tc cuprates.