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The primary cilium, a hair-like projection from the cellular membrane, is involved in fluid flow sensing. Bending the primary cilium is known to trigger several signaling pathways. In this study, the primary cilium is modeled as a thin beam undergoing large deflection due to fluid drag forces. For the first time, the bending response is analyzed with a model combining large angle rotations with the assumption of a linear drag force along the ciliary length. In addition, the initial curvature and the angle between the cell membrane and the cilium are integrated into the model. The model is applied on three dimensional confocal images of fluorescent cilia in living cells that are exposed to laminar shear flow. The coordinates of the cilia are computed by a simple and effective image processing algorithm. By fitting the equations of the model to the coordinates of the bent cilia, the flexural rigidity and the angle between the cilium and the cell membrane are estimated. The flexural rigidity of the primary cilium is approximately ${10}^{-23}$ Nm, but the values vary for different cilia, which suggests that the mechanical properties of the primary cilia are heterogeneous. Interestingly, the base of the cilium doesn't deflect under fluid flow which means that the primary cilium is firmly anchored. Using immunocytochemistry, the connections of the microtubule network to the base of the cilium could be resolved. The microtubules may provide the mechanical stability of the cilium. Incorporating a linear drag force and allowing a basal tilt is an important step towards a more realistic mechanical model of the primary cilium and towards more accurate values of its flexural rigidity. The model together with the imaging technique is a useful tool to study the mechanical behavior of the primary cilium and will eventually lead to new insight in the poorly understood mechanisms of mechanotransduction.