<prompt> Vekta zilizopewa mathbfa na mathbfb, acha mathbfp iwe vekta kiasi kwambamathbfp - mathbfb 2 mathbfp - mathbfa.Kati ya vekta zote kama hizo mathbfp, kuna viasili t na u kiasi kwamba mathbfp iko katika umbali maalum kutoka t mathbfa u mathbfb. Ingiza jozi iliyoagizwa t,u.  <response>: Kutoka mathbfp - mathbfb 2 mathbfp - mathbfa,mathbfp - mathbfb2 4 mathbfp - mathbfa2.Hii inapanuka kamamathbfp2 - 2 mathbfb cdot mathbfp mathbfb2 4 mathbfp 2 - 8 mathbfa cdot mathbfp 4 mathbfa2,ambayo hurahisisha hadi 3 mathbfp2 8 mathbfa cdot mathbfp - 2 mathbfb cdot mathbfp - 4 mathbfa2 mathbfb2 . Kwa hiyo,mathbfp2 frac83 mathbfa cdot mathbfp - frac23 mathbfb cdot mathbfp - frac43 mathbfa2 frac13 mathbfb2.Tunataka mathbfp - t mathbfa u mathbfb kuwa mara kwa mara, ambayo ina maana mathbfp - t mathbfa - u mathbfb2 haibadiliki. Hii inapanuka kamaanzaalignmathbfp - t mathbfa - u mathbfb2 mathbfp2 t2 mathbfa 2 u2 mathbfb2 - 2t mathbfa cdot mathbfp - 2u mathbfb cdot mathbfp 2tu mathbf a cdot mathbfb frac83 mathbfa cdot mathbfp - frac23 mathbfb cdot mathbfp - frac4 3 mathbfa2 frac13 mathbfb2 quad t2 mathbfa2 u2 mathbfb2 - 2t mathbfa cdot mathbfp - 2u mathbfb cdot mathbfp 2tu mathbfa cdot mathbfb kushoto frac83 - 2t kulia mathbfa cdot mathbfp - kushoto frac23 2u kulia mathbf b cdot mathbfp quad kushoto t2 - frac43 kulia mathbfa2 kushoto u2 frac13 kulia mathbfb2 2tu mathbfa cdot mathbfb.endalignMasharti pekee yasiyo ya kudumu katika usemi huu ni left frac83 - 2t right mathbfa cdot mathbfp na left frac23 2u kulia mathbfb cdot mathbfp. Tunaweza kuzifanya kuwa sawa 0 kwa kuweka 2t frac83 na 2u -frac23. Hizi husababisha t frac43 na u -frac13, so t,u kushoto frac43, -frac13 kulia. 