We explain and demonstrate a new method of force and position calibrations for optical tweezers with back-focal-plane photodetection. The method combines power spectral measurements of thermal motion and the response to a sinusoidal motion of a translation stage. It consequently does not use the drag coefficient of the trapped object as an input. Thus, neither the viscosity, nor the size of the trapped object, nor its distance to nearby surfaces needs to be known. The method requires only a low level of instrumentation and can be applied in situ in all spatial dimensions. It is both accurate and precise: true values are returned, with small error bars. We tested this experimentally, near and far from surfaces in the lateral directions. Both position and force calibrations were accurate to within 3%. To calibrate, we moved the sample with a piezoelectric translation stage, but the laser beam could be moved instead, e. g., by acousto-optic deflectors. Near surfaces, this precision requires an improved formula for the hydrodynamical interaction between an infinite plane and a microsphere in nonconstant motion parallel to it. We give such a formula. (c) 2006 American Institute of Physics.