<prompt> Makadirio ya beginpmatrix -8 b endpmatrix kwenye beginpmatrix 2 1 endpmatrix is-frac135 anzapmatrix 2 1 endpmatrix.Tafuta b.  <response>: Ili kupata makadirio ya vekta kwenye vekta nyingine, naweza kutumia fomula textproj_mathbfu mathbfv fracmathbfv cdot mathbfu mathbfu cdot mathbfu mathbfu, ambapo mathbfu ndio vekta ninayoonyesha na mathbfv ndio vekta I m inayoonyesha. Katika kesi hii, mathbfu anzapmatrix 2 1 endpmatrix na mathbfv anzapmatrix -8 b mwishopmatrix. Nikichomeka hizi kwenye fomula, napata textproj_mathbfu mathbfv fracbeginpmatrix -8 b endpmatrix cdot begin pmatrix 2 1 mwishopmatrixanzapmatrix 2 1 mwishopmatrix cdot anzapmatrix 2 1 mwishopmatrix anzapmatrix 2 1 endpmatrix. Kwa kurahisisha bidhaa za nukta, napata textproj_mathbfu mathbfv frac-16 b5 beginpmatrix 2 1 endpmatrix. Sasa, ninahitaji kuweka hii sawa na -frac135 beginpmatrix 2 1 endpmatrix na solve kwa b. Nikilinganisha mgawo wa vekta mbili, ninapata -16 b -13, hivyo b 3. 