<prompt> Wakati vekta mathbfa kuanzapmatrix 1 -1 2 mwishopmatrix na mathbfb anzapmatrix 0 3 0 endpmatrix zote zinakadiriwa kwenye vekta sawa mathbfv, matokeo yake ni mathbfp katika visa vyote viwili. Zaidi ya hayo, vekta mathbfa, mathbfb, na mathbfp ni collinear. Tafuta mathbfp.  <response>: Kwanza, tunapata mstari unaopita mathbfa na mathbfb. Mstari huu unaweza kuainishwa namathbfp anzapmatrix 1 -1 2 mwishopmatrix t kushoto anzapmatrix 0 3 0 mwishopmatrix - startpmatrix 1 -1 2 mwishopmatrix kulia anzapmatrix 1 -1 2 mwishopmatrix t anzapmatrix -1 4 -2 mwishopmatrix anzapmatrix -t 1 4t - 1 -2t 2 mwishopmatrix.asyusepackageamsmath;ukubwa wa kitengo 1 cm;jozi A, B, O, P;A -5,1;B 2,3;O 0,0;P O tafakariA,BO2;choraO--A,Arrow6;choraO--B,Arrow6;choraO--P,Arrow6;kuchorainterpA,B,-0.1--interpA,B,1.1,iliyopigwa;leboanzapmatrix 1 -1 2 mwishopmatrix, A, N;leboanzapmatrix 0 3 0 mwishopmatrix, B, N;lebomathbfp, P, N;asyVekta mathbfp yenyewe itakuwa ya orthogonal kwa mwelekeo wa vekta beginpmatrix -1 4 -2 endpmatrix, hivyoanzapmatrix -t 1 4t - 1 -2t 2 mwishopmatrix cdoti anzapmatrix -1 4 -2 mwishopmatrix 0 .Kwa hivyo, -t 1-1 4t - 14 -2t 2-2 0. Kutatua, tunapata t frac 37. Kwa hivyo, mathbfp anzapmatrix 47 57 87 endpmatrix. 