<prompt> Acha mathbfa, mathbfb, na mathbfc ziwe vivekta vitatu hivi kwamba mathbfa mathbfb mathbfc 2. Pia, pembe kati ya vivekta viwili hivi ni arccos frac58. Tafuta sauti ya parallelepiped inayozalishwa na mathbfa, mathbfb , na mathbfc.  <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 60, 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 4 na mathbfa cdot mathbf b mathbfa cdot mathbfc 2 cdot 2 cdot frac58 frac52. Pia, mathbfa cdot mathbfn 0, hivyofrac52 4s frac5t2.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 frac52 frac5s2 4t.Kutatua milinganyo frac52 4s frac5t2 na frac52 frac5s2 4t, tunapata s t frac513. Kwa hiyo,mathbfp frac513 mathbfa mathbfb.Kishaanzaalignmathbfp2 frac25169 mathbfa cdot mathbfa 2 mathbfa cdot mathbfb mathbfb cdot mathbfb frac25169 kushoto 4 2 cdot frac52 4 kulia frac2513.endalignNa Pythagoras, urefu wa parallelepiped basi hutolewa nasqrt4 - mathbfp2 sqrt4 - frac2513 sqrtfrac2713. Msingi wa bomba la parallelepiped una eneo 2 cdot 2 cdot sin left arccos frac58 kulia 4 sqrt1 - left frac58 kulia2 4 sqrtfrac3964, hivyo ujazo wa parallelepiped nisqrtfrac2713 cdot 4 sqrtfrac3964 frac92. 