<prompt> Acha mathbfa, mathbfb, na mathbfc ziwe vivekta vitatu, hivi kwamba pembe kati ya mojawapo ya hizo ni pembe ya papo hapo theta. Kiasi cha tetrahedron inayozalishwa na vekta hizi tatu ni frac1sqrt360. Find3 cos2 theta - 2 cos3 theta.  <response>: Acha mathbfp iwe makadirio ya mathbfc kwenye ndege iliyo na mathbfa na mathbfb.asykuagiza tatu;ukubwa 140;makadirio ya sasa mtazamo 6,3,2;halisi t 40, k Cos t;mara tatu A, B, C, O, P, Q;A Cost2,Sint2,0;B Cost2,-Sint2,0;C kCost2,0,sqrt1 - k2Cost22;O 0,0,0;P kCost2,0,0;Q kk 1A kk 1B;choraO--A,Arrow36;choraO--B,Arrow36;choraO--C,Arrow36;choraO--P,Arrow36;kuchoraC--P,dashed;lebomathbfa, A, S, fontsize10;lebomathbfb, B, W, fontsize10;lebomathbfc, C, NW, fontsize10;lebomathbfp, P, SW, fontsize10;asyKishamathbfp s mathbfa t mathbfbkwa baadhi ya alama s na t. Acha mathbfn iwe kivekta cha kawaida kwa ndege iliyo na mathbfa na mathbfb, hivyomathbfc mathbfp u mathbfn s mathbfa t mathbfb u mathbfnkwa baadhi ya scalar u.Kuchukua bidhaa ya nukta na mathbfa, tunapatamathbfa cdot mathbfc s mathbfa cdot mathbfa t mathbfa cdot mathbfb u mathbfa cdot mathbfn.Kumbuka kwamba mathbfa cdot mathbfa mathbfa2 1 na mathbfa cdot mathbf b mathbfa cdot mathbfc cos theta. Acha k cos theta, hivyo mathbfa cdot mathbfb mathbf a cdot mathbfc k. Pia, mathbfa cdot mathbfn 0, hivyok s tk.Vile vile, kuchukua bidhaa ya nukta na mathbfb, tunapatamathbfb cdot mathbfc s mathbfa cdot mathbfb t mathbfb cdot mathbfb u mathbfb cdot mathbfn.Hii inapungua hadi k sk t.Kutatua kwa s na t katika mfumo k s tk, k sk t, tunapata s t frackk 1. Kwa hivyo,mathbfp frackk 1 mathbfa mathbfb.Kishaanzaalignmathbfp2 frack2k 12 mathbfa cdot mathbfa 2 mathbfa cdot mathbfb mathbfb cdot mathbfb frack2k 12 1 2k 2 frack2k 12 cdoti 2k 1 frac2k2k 1.endalignNa Pythagoras, urefu wa parallelepiped basi hutolewa nasqrt1 - mathbfp2 sqrt1 - frac2k2k 1 sqrtfrac-2k2 k 1k 1 sqrtfrac2k 11 - k1 k.Kizio cha bomba la parallelepiped kina eneo sin theta sqrt1 - k2 sqrt1 k1 - k, hivyo ujazo wa parallelepiped nisqrtfrac2k 11 - k1 k cdot sqrt1 - k1 k 1 - k sqrt 2k 1.Kiasi cha tetrahedron sambamba basi ni frac16 1 - k sqrt2k 1.Kwa hivyo,frac16 1 - k sqrt2k 1 frac1sqrt360,hivyo 1 - k sqrt2k 1 frac6sqrt360. Kuweka pande zote mbili, tunapata1 - k2 2k 1 frac36360 frac110.Hii inapanuka kama2k3 - 3k2 1 frac110.Kwa hivyo,3 cos2 theta - 2 cos3 theta 3k2 - 2k3 frac910. 