<prompt> Acha mathbfa, mathbfb, na mathbfc ziwe vivekta kiasi kwamba mathbfa mathbfb 1, mathbfc frac2sqrt7, namathbfc mathbfc nyakati mathbfa mathbfb.Tafuta pembe ndogo iwezekanavyo kati ya mathbfa na mathbfc, kwa digrii.  <response>: Tangu mathbfc mathbfc nyakati mathbfa mathbfb,mathbfc mathbfc nyakati mathbfa cdot mathbfc mathbfc nyakati mathbfa mathbfb cdot mathbfb.Hii inapanuka kamamathbfc cdot mathbfc 2 mathbfc cdot mathbfc nyakati mathbfa mathbfc nyakati mathbfa cdot mathbfc nyakati mathbfa mathbfb cdot mathbfb.Tunajua mathbfb cdot mathbfb mathbfb2 1 na mathbfc cdot mathbfc mathbfc2 frac47. Kwa kuwa mathbfc times mathbfa ni orthogonal kwa mathbfc,mathbfc cdot mathbfc nyakati mathbfa 0.Mwishowe, mathbfc mara mathbfa cdot mathbf c nyakati mathbfa mathbfc nyakati mathbfa2. Acha theta iwe pembe kati ya mathbfa na mathbfc. Kishamathbfc nyakati mathbfa mathbfa mathbfc sin theta frac2sqrt7 sin theta,hivyo mathbfc nyakati mathbfa2 frac4 7 dhambi2 theta. Kwa hiyo,frac47 frac47 sin2 theta 1.Hii husababishasin2 theta frac34.hivyosin theta pm fracsqrt32.Pembe ndogo iwezekanavyo theta basi ni 60circ.Vekta mathbfa anzapmatrix 12 sqrt32 0 mwishopmatrix, mathbfb anzapmatrix 2 sqrt7 0 sqrt37 mwishopmatrix, na mathbfc anzapmatrix 2sqrt7 0 0 endpmatrix huonyesha kuwa pembe ya 60circ inaweza kufikiwa. 