<prompt> Wakati beginpmatrix a b endpmatrix inakadiriwa kwenye beginpmatrix sqrt3 1 endpmatrix, vekta inayotokana ina ukubwa sqrt 3. Pia, a 2 b sqrt3. Weka thamani zote zinazowezekana za a, zikitenganishwa na koma.  <response>: Kutoka kwa fomula ya makadirio,jina la operetaproj_anzapmatrix sqrt3 1 mwishopmatrix anzapmatrix a b mwishopmatrix fracanzapmatrix a b mwishopmatrix cdot anzapmatrix sqrt3 1 mwishopmatrixleft anzapmatrix sqrt3 1 mwishopmatrix kulia2 anzapmatrix sqrt3 1 mwishopmatrix fraca sqrt 3 b 4 anzapmatrix sqrt3 1 endpmatrix.Vekta hii ina ukubwakushoto fraca sqrt3 b4 anzapmatrix sqrt3 1 mwishopmatrix kulia fraca sqrt3 b4 kushoto anzapmatrix sqrt3 1 mwishopmatrix kulia fraca sqrt3 b4 cdot 2 fraca sqrt3 b2.Hivyo, tunataka frac a sqrt3 b2 sqrt3. Sawa, a sqrt3 b 2 sqrt3, au a sqrt3 b2 12.Pia, a 2 b sqrt3, hivyo2 sqrt3 4b2 12.Kisha 2 sqrt3 4b pm 2 sqrt3. Hii inaleta suluhu b -sqrt 3 na b 0, ambayo nayo inaongoza kwa thamani a -1 na a boxed2. 