<prompt> Matrix ya kuonyesha kwenye ndege fulani P, ambayo hupitia asili, imetolewa nanewcommandarraystretch1.5 anzapmatrix frac1314 -frac17 frac314 -frac1 7 frac57 frac37 frac314 frac37 frac514 endpmatrix renewcommandarraystretch1.Tafuta vekta ya kawaida ya ndege P. Weka jibu lako kwa njia ya beginpmatrix a b c endpmatrix , ambapo a, b, na c ni nambari kamili, a 0, na gcda,b,c 1.  <response>: Kwa kuwa beginpmatrix a b c endpmatrix ni vekta ya kawaida, makadirio yake ni vekta sifuri. Hivyo,newcommandarraystretch1.5 anzapmatrix frac1314 -frac17 frac314 -frac1 7 frac57 frac37 frac314 frac37 frac514 endpmatrix newcommandarraystretch1 anzapmatrix a b c endpmatrix anzapmatrix 0 0 0 mwishopmatrix. Kisha frac1314 a - frac17 b frac314 0, -frac17 a frac 57 b frac37 c 0, na frac314 a frac37 b frac514 0. Hizi hupunguza hadianzaalign13a - 2b 3c 0, -a 5b 3c 0, 3a 6b 5c 0.endalignTukiondoa milinganyo miwili ya kwanza, tunapata 14a - 7b 0, hivyo b 2a. Kisha-a 10a 3c 0,hivyo c -3a. Kwa hiyo,anzapmatrix a b c mwishopmatrix anzapmatrix a 2a -3a mwishopmatrix a anzapmatrix 1 2 -3 mwishopmatrix,kwa hivyo vekta tunayotafuta ni beginpmatrix 1 2 -3 endpmatrix. 