<prompt> Acha P iwe ndege inayopitia asili na vekta ya kawaida anzapmatrix 1 -2 1 endpmatrix. Tafuta matrix mathbfP ambayo kwa vekta yoyote. mathbfv, mathbfP mathbfv ni makadirio ya mathbfv kwenye ndege P.  <response>: Acha mathbfv anzepmatrix x y z endpmatrix, na mathbfp iwe makadirio ya mathbfp kwenye ndege P. Kisha mathbfv - mathbfp ni makadirio ya mathbfv kwenye vekta ya kawaida mathbfn anzapmatrix 1 -2 1 mwishopmatrix.asykuagiza tatu;ukubwa 160;makadirio ya sasa mtazamo 6,3,2;mara tatu I 1,0,0, J 0,1,0, K 0,0,1;mara tatu O 0,-0.5,0, V 0,1.5,1, P 0,1.5,0;chorauso2I 2J--2I - 2J---2I - 2J---2I 2J-- mzunguko, manjano isiyokolea, hakuna mwanga;chora2I 2J--2I - 2J---2I - 2J---2I 2J--mzunguko ;chora P 0.1O - P--P 0.1O - P 0.2V - P--P 0.2V - P;choraO--P,kijani,Arrow36;choraO--V,nyekundu,Arrow36;choraP--V,bluu,Arrow36;chora1,-0.8,0--1,-0.8,0.2--1,-1,0.2;chora1,-1,0--1,-1,2,majenta,Mshale36;lebomathbfv, V, N, fontsize10;lebomathbfp, P, S, fontsize10;lebomathbfn, 1,-1,1, dir180, fontsize10;lebomathbfv - mathbfp, V P2, E, fontsize10;asyHivyo,mathbfv - mathbfp fracanzapmatrix x y z mwishopmatrix cdot anzapmatrix 1 -2 1 mwishopmatrixanzapmatrix 1 -2 1 mwishopmatrix cdot anzapmatrix 1 -2 1 mwishopmatrix anzapmatrix 1 -2 1 mwishopmatrix fracx - 2y z6 anzapmatrix 1 -2 1 endpmatrix newcommandarraystretch 1.5 anzapmatrix frac16 x - frac13 y frac16 z -frac13 x frac23 y - frac13 z frac16 x - frac13 y frac16 z endpmatrix newcommandarraystretch1.Kishamathbfp anzapmatrix x y z mwishopmatrix - upya amriarraystretch1.5 anzapmatrix frac16 x - frac13 y frac16 z -frac13 x frac23 y - frac13 z frac16 x - frac13 y frac16 z endpmatrix renewcommandarraystretch1 newcommand arraystretch1.5 anzapmatrix frac56 x frac13 y - frac16 z frac13 x frac13 y frac13 z -frac16 x frac13 y frac5101 6 z endpmatrix newcommandarraystretch1 renewcommandarraystretch1.5 beginpmatrix frac56 frac13 -frac16 frac13 frac13 frac13 -frac16 frac13 frac56 endpmatrix renewcommandarraystretch1 beginpmatrix x y z endpmatrix. Kwa hivyo,mathbfP anzapmatrix frac56 frac13 -frac16 frac13 frac13 frac13 -frac16 frac13 frac56 end PMmatrix. 