<prompt> Vekta mathbfa, mathbfb, na mathbfc inatosheleza mathbfa mathbfb 1, mathbfc 2, namathbfa nyakati mathbfa nyakati mathbfc mathbfb mathbf0.Kama theta ni pembe kati ya mathbf a na mathbfc, kisha utafute thamani zote zinazowezekana za theta, kwa digrii.  <response>: Suluhisho 1. Kwa bidhaa ya vekta tatu, mathbfu mara mathbfv mara mathbfw mathbfu cdot mathbfw mathbfv - mathbfu cdot mathbfv mathbfw, hivyomathbfa cdot mathbfc mathbfa - mathbfa cdot mathbfa mathbfc mathbfb mathbf 0.Kwa kuwa mathbfa cdot mathbfa mathbfa2 1, hii inatuambiamathbfc mathbfa cdot mathbfc mathbfa mathbfb.Hebu k mathbfa cdot mathbf c, hivyo mathbfc k mathbfa mathbfb. Kishamathbfc2 k mathbfa mathbfb2.Tangu mathbfb -mathbfa nyakati mathbfa nyakati mathbfc, vekta mathbfa na mathbfb ni za orthogonal. Kwa hivyo,4 k2 1,hivyo k pm sqrt3. Kishacos theta fracmathbfa cdot mathbfcmathbfa mathbfc pm fracsqrt32,ili theta inaweza kuwa 30circ au boxed150circ .Suluhisho 2. Bila kupoteza kwa ujumla, tunaweza kudhani kuwa mathbfa anzapmatrix 0 0 1 endpmatrix. Hebu mathbfc anzapmatrix x y z mwishopmatrix. Kishamathbfa nyakati mathbfa nyakati mathbfc mathbfa nyakati anzapmatrix -y x 0 mwishopmatrix anzapmatrix -x -y 0 mwishopmatrix,hivyo mathbfb anzapmatrix x y 0 mwishopmatrix.Tangu mathbfb 1 na mathbfc 2, x2 y2 1 na x2 y2 z2 4. Kwa hivyo, z2 3, hivyocos theta fracmathbfa cdot mathbfcmathbfa mathbfc fracz2 pm fracsqrt32.Hii inamaanisha kuwa thamani zinazowezekana za theta ni 30 circ au boxed150circ. 