Lorentz force formula F=qvB has been obtained by experiment. It can be derived from third Hertz's equation:
rot E = -d/dt B - rot(v x B)
if derived in such way it changes sign, F=-qvB. Another two formulas can be derived from Lorentz force formula:
U=BLv; induced voltage equals induction times length times speed F=BIL; mechanical force equals induction times current times length
both change sign, i.e. U=-BLv and F=-BIL.