<prompt> Acha mathbfa na mathbfb ziwe vekta za orthogonal. Ikiwa operatornameproj_mathbfa anzapmatrix 3 -3 endpmatrix anzapmatrix -frac35 -frac 65 endpmatrix, kisha tafuta operatornameproj_mathbfb beginpmatrix 3 -3 endpmatrix.  <response>: Tangu beginpmatrix -frac35 -frac65 endpmatrix ni makadirio ya beginpmatrix 3 -3 end pmatrix kwenye mathbfa,anzapmatrix 3 -3 mwishopmatrix - anzapmatrix -frac35 -frac65 endpmatrix startpmatrix frac185 -frac95 endpmatrixni ya orthogonal kwa mathbfa. Lakini kwa kuwa mathbfa na mathbfb ni orthogonal, beginpmatrix frac185 -frac95 endpmatrix ni kizidishio cha scalar cha mathbf b.asyusepackageamsmath;ukubwa wa kitengo 1 cm;jozi A, B, O, P, Q, V;A 1,2;B 2,-1;O 0,0;V 3,-3;P V tafakariO,AV2;choraO--V,Arrow6;choraO--P,Arrow6;choraP--V,Arrow6;chora-1,0--4,0;chora0,-4--0,1;leboanzapmatrix 3 -3 mwishopmatrix, V, SE;lebobeginpmatrix -frac35 -frac65 endpmatrix, P, W;asyZaidi ya hayo,anzapmatrix 3 -3 mwishopmatrix - anzapmatrix frac185 -frac95 endpmatrix anza pmatrix -frac35 -frac65 endpmatrixni kizidishio cha mathbfa, na kwa hivyo ni orthogonal hadi mathbf b. Kwa hivyo, operatornameproj_mathbfb anzapmatrix 3 -3 endpmatrix anzapmatrix frac185 -frac95 endpmatrix. 