By using a high quality ${\text{Tl}}_{2}{\text{Ba}}_{2}{\text{Ca}}_{2}{\text{Cu}}_{3}{\text{O}}_{10}$ (Tl-2223) single crystal as an example, the magnetization vector was probed in the reversible region of highly anisotropic cuprate superconductors. For that, we have measured its components along and transverse to the applied magnetic field for different crystal orientations. The analysis shows that the angular dependence of the perpendicular component of the magnetization vector follows the one predicted by a London-type approach which includes a contribution associated with the thermal fluctuations of the two-dimensional vortex positions. For the Tl-2223 crystal studied here, a lower bound for the anisotropy factor was estimated to be about 190.