<prompt> Acha theta iwe pembe kati ya mstarifracx 12 fracy3 fracz - 36na ndege -10x - 2y 11z 3. Find dhambi theta.asykuagiza tatu;ukubwa 150;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;chorauso2I 2J--2I - 2J---2I - 2J---2I 2J-- mzunguko, manjano isiyokolea, hakuna mwanga;chora2I 2J--2I - 2J---2I - 2J---2I 2J--mzunguko ;kuchora0,0,0---0.5,1.5,1;chora0,0,0--0.8-0.5,1.5,1,Arrow36;chora0,0,0--1.2-0.5,-1.5,-1, iliyokatika;kuchora1.2-0.5,-1.5,-1--2-0.5,-1.5,-1;kuchora0,0,0---0.5,1.5,0;lebotheta, 0.5-0.5,1.5,0.0 0,0,0.3;nukta 0,0,0;asy  <response>: Vekta ya mwelekeo wa mstari ni mathbfd anzapmatrix 2 3 6 endpmatrix, na vekta ya kawaida kwa ndege ni mathbfn anza pmatrix -10 -2 11 endpmatrix. Kumbuka kwamba ikiwa theta ni pembe kati ya mathbfd katika ndege, basi pembe kati ya mathbfd na mathbfn ni 90circ - theta.asykuagiza tatu;ukubwa 150;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;chorauso2I 2J--2I - 2J---2I - 2J---2I 2J-- mzunguko, manjano isiyokolea, hakuna mwanga;chora2I 2J--2I - 2J---2I - 2J---2I 2J--mzunguko ;kuchora0,0,0---0.5,1.5,1;chora0,0,0--0.8-0.5,1.5,1,Arrow36;chora0,0,0--1.2-0.5,-1.5,-1, iliyokatika;kuchora1.2-0.5,-1.5,-1--2-0.5,-1.5,-1;kuchora0,0,0---0.5,1.5,0;chora0,0,0--0,0,1,Arrow36;lebotheta, 0.5-0.5,1.5,0.0 0,0,0.3;lebomathbfd, -0.5,1.5,1, NE;lebomathbfn, 0,0,1, N;nukta 0,0,0;asyKwa hiyo,cos 90circ - theta fracmathbfd cdot mathbfnmathbfd mathbfn fracanzapmatrix 2 3 6 mwishopmatrix cdot anzapmatrix -10 -2 11 mwisho pmatrixleft anzapmatrix 2 3 6 mwishopmatrix kulia kushoto anzapmatrix -10 -2 11 mwishopmatrix kulia frac407 cdot 15 frac821.Kwa hiyo, sin theta frac821. 