<prompt> Acha P iwe kigezo katika nafasi ya kuratibu, ambapo viwianishi vyote vya P ni vyema. Mstari kati ya asili na P umechorwa. Pembe kati ya mstari huu na mhimili wa x-, y-, na z-axis ni alpha,beta, na gamma, mtawalia. Ikiwa cos alpha frac13 na cos beta frac15, basi bainisha cos gamma.asykuagiza tatu;ukubwa 180;makadirio ya sasa mtazamo 6,3,2;mara tatu I 1,0,0, J 0,1,0, K 0,0,1, O 0,0,0;mara tatu V 3,2,2, P;P 2.5I 2.5VabsV2;kuchora1.1I..1.5PabsP..1.5VabsV;leboalpha, 1.5PabsP, NW;P 2.5J 2.5VabsV2;kuchora1.5J..1.5PabsP..1.5VabsV;lebobeta, 1.5PabsP, NE;P 2.5K 2.5VabsV2;chora1.5K..1.5PabsP..1.5VabsV;lebogamma, 1.5PabsP, E;kuchoraO--5.5VabsV;kuchoraO--3I, Arrow36;choraO--3J, Arrow36;choraO--3K, Arrow36;lebox, 3.2I;leboy, 3.2J;leboz, 3.2K;nukta P, 5.5VabsV, NE;asy  <response>: Acha O iwe asili, na acha P x,y,z. Acha X iwe mguu wa perpendicular kutoka P hadi x-mhimili. Kisha angle POX alpha, OP sqrtx2 y2 z2, na OX x, hivyocos alpha fracxsqrtx2 y2 z2.asyukubwa wa kitengo 1 cm;chora0,0--3,0--3,2--mzunguko;leboP x,y,z, 3,2, NE;lebox, 3,1, E, nyekundu;lebosqrtx2 y2 z2, 32,1, NW, nyekundu;leboalpha, 0.9,0.3;leboO, 0,0, SW;leboX, 3,0, SE;asyVile vile, cos beta fracysqrtx2 y2 z2 na cos gamma fraczsqrtx2 y2 z2. Kwa hivyo,cos2 alpha cos2 beta cos2 gamma 1.Kwa kuwa cos alpha frac13 na cos beta frac15,cos2 gamma 1 - cos2 alpha - cos2 beta frac191225.Kwa kuwa gamma ni kali, cos gamma fracsqrt19115. 