<prompt> Wakati vekta anzapmatrix -5 1 endpmatrix na beginpmatrix 2 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 -5 1 endpmatrix na beginpmatrix 2 3 endpmatrix . Mstari huu unaweza kurekebishwa naanzapmatrix -5 1 mwishopmatrix t kushoto anzapmatrix 2 3 mwishopmatrix - anzapmatrix -5 1 mwisho pmatrix kulia anzapmatrix -5 1 mwishopmatrix t anzapmatrix 7 2 mwishopmatrix anzapmatrix 7t - 5 2t 1 mwishopmmatrix.asyusepackageamsmath;ukubwa wa kitengo 1 cm;jozi A, B, O, P;A -5,1;B 2,3;O 0,0;P O tafakariA,BO2;chora-6,0--3,0;chora0,-1--0,4;choraO--A,Arrow6;choraO--B,Arrow6;choraO--P,Arrow6;kuchorainterpA,B,-0.1--interpA,B,1.1,iliyopigwa;leboanzapmatrix -5 1 mwishopmatrix, A, N;leboanzapmatrix 2 3 mwishopmatrix, B, N;lebomathbfp, P, N;asyVekta mathbfp yenyewe itakuwa ya orthogonal kwa vekta ya mwelekeo beginpmatrix 7 2 endpmatrix, hivyoanzapmatrix 7t - 5 2t 1 mwishopmatrix cdot anzapmatrix 7 2 mwishopmatrix 0.Kwa hiyo, 7t - 5 7 2t 12 0. Kutatua, tunapata t frac3353. Kwa hivyo, mathbfp beginpmatrix -3453 11953 mwishopmatrix. 