<prompt> Katika pembetatu ABC, D iko kwenye overlineBC iliyopanuliwa zaidi ya C kiasi kwamba BD:DC 3:1, na E iko kwenye overlineAC hivi kwamba AE:EC 5:3. Acha P iwe makutano ya mistari BE na AD.asyukubwa wa kitengo sentimita 0.8;jozi A, B, C, D, E, F, P;A 1,4;B 0,0;C 6,0;D interpB,C,32;E interpA,C,58;P kiendeleziA,D,B,E;kuchoraA--B--C--mzunguko;kuchoraA--D--C;kuchoraB--P;leboA, A, N;leboB, B, SW;leboC, C, S;leboD, D, SE;leboE, E, S;leboP, P, NE;asyKishaarrow overrightarrowP x overrightarrowA y overrightarrowB z overrightarrowC,ambapo x, y, na z ni vitu visivyobadilika hivi kwamba x y z 1. Ingiza x,y,z iliyoagizwa mara tatu.  <response>: Kutoka kwa habari iliyotolewa,fracoverrightarrowD - overrightarrowB3 overrightarrowD - overrightarrowC.Kutenga overrightarrowD, tunapataoverrightarrowD frac32 overrightarrowC - frac12 overrightarrowB.Pia,overrightarrowE frac38 overrightarrowA frac58 overrightarrowC.Kutenga overrightarrowC katika kila mlingano, tunapataoverrightarrowC frac2 overrightarrowD overrightarrowB3 frac8 overrightarrowE - 3 overrightarrowA5. Kisha 10 overrightarrowD 5 overrightarrowB 24 overrightarrowE - 9 overrightarrowA, hivyo 10 overrightarrowD 9 overrightarrowA 24 overrightarrow E - 5 mshale wa kulia zaidiB, aufrac1019 overrightarrowD frac919 overrightarrowA frac2419 overrightarrowE - frac5 19 mshale uliopitilizaB.Kwa kuwa viambatanishi vya pande zote mbili za mlingano vinaongeza hadi 1, vekta iliyo upande wa kushoto iko kwenye mstari AD, na vekta iliyo upande wa kulia iko kwenye mstari BE. Kwa hivyo, vekta hii ya kawaida ni overrightarrowP. KishaanzaalignoverrightarrowP frac1019 overrightarrowD frac919 overrightarrowA frac1019 kushoto frac32 overrightarrowC - frac12 overrightarrowB kulia frac9101 19 mshale wa kulia zaidiA frac919 overrightarrowA - frac519 overrightarrowB frac1519 overrightarrowC.endalignHivyo, x,y,z left frac919, -frac519, frac1519 kulia . 