<prompt> Acha mathbfa, mathbfb, na mathbfc ziwe vekta zisizo na zero, hakuna mbili zinazolingana, ilimathbfa nyakati mathbfb nyakati mathbfc frac13 mathbfb mathbfc mathbfa.Acha theta iwe pembe kati ya mathbfb na mathbfc. Tafuta sin theta.  <response>: Kwa bidhaa ya vekta tatu, kwa vekta zozote mathbfp, mathbfq, na mathbfr,mathbfp nyakati mathbfq mara mathbfr mathbfp cdot mathbfr mathbfq - mathbfp cdot mathbfq mathbfr.Hivyo, mathbfa nyakati mathbfb nyakati mathbfc -mathbfc nyakati mathbfa nyakati mathbfb - mathbfb cdot mathbfc mathbfa mathbfa cdot mathbfc mathbfb. Kwa hiyo,mathbfa cdot mathbfc mathbfb - mathbfb cdot mathbfc mathbfa frac13 mathbfb mathbfc mathbfa.Kishamathbfa cdot mathbfc mathbfb left mathbfb cdot mathbfc frac13 mathbf b mathbfc kulia mathbfa.Kwa kuwa viwekaji data vya mathbfa na mathbfb havilingani, njia pekee ambayo equation hapo juu inaweza kushikilia ni ikiwa pande zote mbili ni sawa na vekta sifuri. Kwa hivyo,mathbfb cdot mathbfc frac13 mathbfb mathbfc 0.Kwa kuwa mathbfb cdot mathbfc mathbfb mathbfc cos theta,mathbfb mathbfc cos theta frac13 mathbfb mathbfc 0.Kwa kuwa mathbfb na mathbfc ni nonzero, inafuata kwamba cos theta -frac13. Kishasin theta sqrt1 - cos2 theta frac2 sqrt23. 