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We examine the effects of local turbulence Reynolds number Rλ and inflow conditions on the magnitude of vorticity in plane turbulent wakes. Measurements of the spanwise component (ω3) of the fluctuating vorticity vector ω = ω1i + ω2j + ω3k (here the subscripts 1, 2 and 3 denote the streamwise, lateral and spanwise directions, respectively) are made in turbulent wakes of a screen and a circular cylinder. Lateral distributions of ω3* (normalized) in general depend on both Rλ and inflow conditions. In the developing region, as the downstream distance x1 increases, ω3* increases significantly in the screen wake but decreases slightly in the cylinder wake. Far downstream in the self-preserving region, ω3* increases linearly with Rλ while it no longer varies with x1 and depends weakly on the influence of inflow conditions. Our analysis suggests that these findings from the measurement of ω3* should apply for the magnitude of ω.