<prompt> Wakati vekta anzapmatrix 4 1 endpmatrix na beginpmatrix -1 3 endpmatrix zote zinakadiriwa kwenye vekta sawa mathbfv , matokeo yake ni mathbfp katika visa vyote viwili. Tafuta mathbfp.  <response>: Kumbuka kuwa vekta mathbfp lazima ilale kwenye mstari unaopita beginpmatrix 4 1 endpmatrix na beginpmatrix -1 3 endpmatrix . Mstari huu unaweza kurekebishwa naanzapmatrix 4 1 mwishopmatrix t kushoto anzapmatrix -1 3 mwishopmatrix - anzapmatrix 4 1 mwishopmatrix kulia anzapmatrix 4 1 mwishopmatrix t anzapmatrix -5 2 mwishopmatrix anzapmatrix -5t 4 2t 1 mwishopmmatrix.asyusepackageamsmath;ukubwa wa kitengo 1 cm;jozi A, B, O, P;A 4,1;B -1,3;O 0,0;P O tafakariA,BO2;chora-2,0--5,0;chora0,-1--0,4;choraO--A,Arrow6;choraO--B,Arrow6;choraO--P,Arrow6;kuchorainterpA,B,-0.1--interpA,B,1.1,iliyopigwa;leboanzapmatrix 4 1 mwishopmatrix, A, N;leboanzapmatrix -1 3 mwishopmatrix, B, N;lebomathbfp, P, N;asyVekta mathbfp yenyewe itakuwa ya orthogonal kwa mwelekeo wa vekta beginpmatrix -5 2 endpmatrix, hivyoanzapmatrix -5t 4 2t 1 mwishopmatrix cdot anzapmatrix -5 2 mwishopmatrix 0.Kwa hiyo, -5t 4-5 2t 12 0. Kutatua, tunapata t frac1829. Kwa hivyo, mathbfp beginpmatrix 2629 6529 mwishopmatrix. 