The Lorentz transformations:

ds = (ds' + bdt')(1-b²)^-1/2

dt = (dt' + bds')(1-b²)^-1/2

ds' = (ds - bdt)(1-b²)^-1/2

dt' = (dt - bds)(1-b²)^-1/2

In the transformations b = ds_r/dt_r is the velocity between the unprimed and the primed Lorentz reference frames, where ds_r is measured in the same direction as ds. In contrast the direction of ds' is reflected so that ds_r = -ds'_r (numerically only) and, hence, -ds_r'/dt'_r = -b. However, measurements may be recorded as positive by applying the same convention in both frames, hence, ds' and ds'_r would not be assigned a minus sign. Either way will not affect the results of the transformation from one hyperbolic vector space to another provided signs are consistently applied to all measurements and the results interpreted accordingly.