<prompt> Acha P iwe ndege inayopitia asili na vekta ya kawaida anzapmatrix 1 1 -1 endpmatrix. Tafuta matrix mathbfR ambayo kwa vekta yoyote. mathbfv, mathbfR mathbfv ni kiakisi cha mathbfv kupitia 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 1 -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 1 -1 mwishopmatrixanzapmatrix 1 1 -1 mwishopmatrix cdot anzapmatrix 1 1 -1 mwishopmatrix anzapmatrix 1 1 -1 mwishopmatrix fracx y - z3 anzapmatrix 1 1 -1 endpmatrix newcommandarraystretch 1.5 anzapmatrix fracx y - z3 fracx y - z3 -fracx y - z3 endpmmatrix renewcommandarraystretch1.Kishamathbfp anzapmatrix x y z mwishopmatrix - upya amriarraystretch1.5 anzapmatrix fracx y - z 3 fracx y - z3 -fracx y - z3 endpmatrix newcommandarraystretch1 mpya amri arraystretch1.5 anzapmatrix frac2x - y z3 frac-x 2y z3 fracx y 2z3 endpmatrix renewcommandarraystretch1.Sasa, acha mathbfr iwe kielelezo cha mathbfv kupitia ndege P.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, R 0,1.5,-1;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;kuchoraO--R,dashed,Arrow36;kuchoraR--P,dashed;lebomathbfv, V, N, fontsize10;lebomathbfp, P, E, fontsize10;lebomathbfn, 1,-1,1, dir180, fontsize10;lebomathbfv - mathbfp, V P2, E, fontsize10;lebomathbfr, R, S;asyKisha mathbfp ndio katikati ya mathbfv na mathbfr, somathbfp fracmathbfv mathbfr2.Tunaweza kutatua kwa mathbfr, kupata mathbfr 2 mathbfp - mathbfv. Kishamathbfr 2 amri upyaarraystretch1.5 anzapmatrix frac2x - y z3 frac-x 2y z3 fracx y 2z3 mwishopmatrix newcommandarraystretch1 - beginpmatrix x y z endpmatrix amri upya arraystretch1.5 anzapmatrix fracx - 2y 2z3 frac-2x y 2z3 frac2x 2y z 3 endpmatrix newcommandarraystretch1 newcommandarraystretch1.5 beginpmatrix frac13 -frac23 frac23 -frac23 frac13 frac23 frac23 frac 23 frac13 endpmatrix renewcommandarraystretch1 beginpmatrix x y z endpmatrix.Hivyo basi ,mathbfR anzapmatrix frac13 -frac23 frac23 -frac23 frac13 frac23 frac23 frac23 frac13 end PMmatrix. 