<prompt> Ikiwa mathbfa, mathbfb, na mathbfc ni vekta kiasi kwamba mathbfa mathbfb 1, mathbfa mathbfb sqrt3, namathbfc - mathbfa - 2 mathbfb 3 mathbfa nyakati mathbfb,kisha tafuta mathbfb cdot mathbfc.  <response>: Ili kupata mathbfb cdot mathbfc, ninahitaji kueleza mathbfc kulingana na mathbfa na mathbfb. ninaweza kutumia equation iliyotolewa mathbfc - mathbfa - 2 mathbfb 3 mathbfa nyakati mathbfb kutenga mathbfc. Kuongeza mathbfa 2 mathbfb kwa pande zote mbili, napata mathbfc mathbfa 2 mathbfb 3 mathbfa nyakati mathbfb. Sasa ninaweza kubadilisha usemi huu kwa mathbfc hadi mathbfb cdot mathbfc na kurahisisha kutumia sifa za bidhaa ya nukta na bidhaa mtambuka. Napata mathbfb cdot mathbfc mathbfb cdot mathbfa 2 mathbfb 3 mathbfa nyakati mathbfb mathbfb cdot mathbfa 2 mathbfb cdot mathbfb 3 mathbfb cdot mathbfa nyakati mathbf b. Bidhaa ya nukta mathbfb na mathbfb ni mathbfb2 12 1. tu mathbfb2 12 1. Bidhaa ya nukta mathbfb na mathbfa nyakati mathbfb ni sifuri, kwa kuwa mathbfa nyakati mathbfb ni sawa na mathbfa na mathbfb. Ili niweze kurahisisha zaidi mathbfb cdot mathbfc mathbfb cdot mathbfa 2. Ili kupata mathbfb cdot mathbfa, Naweza kutumia ukweli kwamba mathbfa mathbfb sqrt3 na fomula ya kawaida ya jumla ya vekta. Napata mathbfa mathbfb2 mathbfa mathbfb cdot mathbfa mathbfb mathbfa cdot mathbfa 2 mathbfa cdot mathbfb mathbfb cdot mathbfb mathbfa 2 2 mathbfa cdot mathbfb mathbfb2 1 2 mathbfa cdot mathbfb 1 2 2 mathbfa cdot mathbfb. Inalinganisha hii na sqrt32 3, napata 2 mathbfa cdot mathbfb 1. Kugawanya kwa 2 , napata mathbfa cdot mathbfb frac12. Kubadilisha hii hadi mathbfb cdot mathbfc, Ninapata mathbf b cdot mathbfc frac12 2 frac52. 