<prompt> Matrix ya kuakisi juu ya mstari fulani ell, ambayo hupitia asili, imetolewa naamri upyaarraystretch1.5 anzapmatrix -frac23 -frac23 -frac13 -frac 23 frac13 frac23 -frac13 frac23 -frac2 3 endpmatrix renewcommandarraystretch1.Tafuta vekta ya mwelekeo wa mstari ell. Ingiza jibu lako katika fomu 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 kweli iko kwenye ell, tafakari huchukua vekta hii yenyewe. Kishaamri upyaarraystretch1.5 anzapmatrix -frac23 -frac23 -frac13 -frac 23 frac13 frac23 -frac13 frac23 -frac2 3 endpmatrix newcommandarraystretch1 anzapmatrix a b c endpmatrix anzapmatrix a b c mwisho pmatrix.Hii inatupaamri upyaarraystretch1.5 anzapmatrix -frac23 a - frac23 b - frac13 c - frac23 a frac13 b frac23 c -frac13 a frac23 b - frac23 c endpmatrix renewcommandarraystretch1 beginpmatrix a b c endpmatrix.Kisha -frac 23 a - frac23 b - frac13 c a, -frac23 a frac13 b frac23 c b, na -frac13 a frac23 b - frac23 c c . Hizi hupunguza hadianzaalign5a 2b c 0, a b - c 0, a - 2b 5c 0.endalignTukiongeza milinganyo miwili ya kwanza, tunapata 6a 3b 0, hivyo b -2a. Kishaa - 2a - c 0,hivyo c -a. Na ikiwa b -2a na c -a, basi mlinganyo wa tatu a - 2b 5c 0 imeridhika. Kwa hivyo,anzapmatrix a b c mwishopmatrix anzapmatrix a -2a -a mwishopmatrix a anzapmatrix 1 - 2 -1 mwishopmatrix.Kwa hivyo, vekta tunayotafuta ni anzapmatrix 1 -2 -1 mwishopmatrix. 