<prompt> Acha mathbfa, mathbfb, mathbfc ziwe vekta kiasi kwamba mathbfa 2, mathbfb 3, namathbfc nyakati mathbfa mathbfb.Tafuta thamani ndogo kabisa ya mathbfc - mathbfa.  <response>: Acha theta iwe pembe kati ya mathbfa na mathbfc, hivyomathbfc nyakati mathbfa mathbfa mathbfc sin theta.Kisha 3 2 mathbfc sin theta, hivyo mathbfc frac32 sin theta.Kwa hivyo,anzaalignmathbfc - mathbfa2 mathbfc2 - 2 mathbfa cdot mathbfc mathbf a2 frac94 sin2 theta - 2 mathbfa mathbfc cos theta 4 frac94 sin2 theta - 2 cdot 2 cdot frac32 sin theta cdot cos theta 4 frac94 sin2 theta - frac6 cos thetasin theta 4.endalignTunaweza kueleza hili kulingana na cot theta:anzaalignfrac94 sin2 theta - frac6 cos thetasin theta 4 frac9 sin2 theta cos2 theta4 sin2 theta - 6 cot theta 4 frac94 frac94 cot2 theta - 6 cot theta 4 frac94 cot2 theta - 6 cot theta frac254.endalignKukamilisha mraba katika cot theta, tunapatamathbfc - mathbfa2 kushoto frac32 cot theta - 2 kulia2 frac94 .Kwa hivyo, thamani ndogo kabisa ya mathbfc - mathbfa ni frac32, ambayo inafikiwa wakati cot theta frac43, au tan theta frac34. 