<prompt> Mzunguko unaozingatia asili huchukua beginpmatrix -4 7 endpmatrix hadi beginpmatrix 1 8 endpmatrix. Mzunguko huchukua vekta gani startpmatrix -1 3 mwishopmatrix hadi?  <response>: Tumbo la mzunguko lazima liwe la umbo beginpmatrix cos theta -sin theta sin theta cos theta endpmatrix. Hivyo,anzapmatrix cos theta -sin theta dhambi theta cos theta mwishopmatrix anzapmatrix -4 7 mwishopmatrix startpmatrix 1 8 mwishopmatrix.Hii inatupa milinganyo -4 cos theta - 7 sin theta 1 na -4 sin theta 7 cos theta 8. Kutatua mfumo huu, tunapata cos theta frac45 na sin theta -frac35. Hivyo, begin pmatrix -1 3 endpmatrix inachukuliwa hadianzapmatrix frac45 frac35 -frac35 frac45 endpmatrix startpmatrix -1 3 mwishopmatrix anzapmatrix 1 3 mwishopmatrix. 