<prompt> Tafuta makadirio ya anzapmatrix 1 2 3 mwishopmatrix kwenye ndege 3x - y 4z 0.  <response>: Acha P iwe ndege 3x - y 4z 0. Tunaweza kuchukua mathbfn beginpmatrix 3 -1 4 endpmatrix kama vekta ya kawaida ya ndege P.Acha mathbfv anzepmatrix 1 2 3 endpmatrix, na mathbfp iwe makadirio yake kwenye ndege P. Kumbuka kuwa mathbf v - mathbfp ni sambamba na mathbfn.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;asyKwa hivyo, mathbfv - mathbfp ni makadirio ya mathbfv kwenye mathbfn. Kwa hivyo,mathbfv - mathbfp fracanzapmatrix 1 2 3 mwishopmatrix cdot anzapmatrix 3 -1 4 mwishopmatrixanzapmatrix 3 -1 4 mwishopmatrix cdot anzapmatrix 3 -1 4 mwishopmatrix anzapmatrix 3 -1 4 mwishopmatrix frac1326 anzapmatrix 3 -1 4 endpmatrix anzapmatrix 32 -12 2 mwishopmmatrix.Kishamathbfp mathbfv - anzapmatrix 32 -12 2 mwishopmatrix anzapmatrix -12 5 2 1 mwishopmmatrix. 