<prompt> Kuna vekta mathbfa na mathbfb vilemathbfa mathbfb anzapmatrix 2 -1 3 mwishopmatrix,ambapo mathbfa ni sambamba na begin pmatrix 1 2 -1 mwishopmatrix, na mathbfb ni orthogonal kwa beginpmatrix 1 2 -1 endpmatrix. Find mathbfb.  <response>: Kwa kuwa mathbfa ni sambamba na beginpmatrix 1 2 -1 endpmatrix,mathbfa t anzapmatrix 1 2 -1 mwishopmatrix anzapmatrix t 2t -t mwishopmatrixkwa baadhi scalar t. Kishamathbfb anzapmatrix 2 -1 3 mwishopmatrix - anzapmatrix t 2t -t mwishopmatrix anzapmatrix 2 - t -1 - 2t 3 t mwishopmatrix.Tunataka hii iwe orthogonal ili anzapmatrix 1 2 -1 mwishopmatrix, hivyoanzapmatrix 2 - t -1 - 2t 3 t mwishopmatrix cdot anzapmatrix 1 2 -1 mwishopmatrix 0. Kisha 2 - t1 -1 - 2t2 3 t-1 0. Kutatua, tunapata t -frac12 . Kisha mathbfb anzapmatrix 52 0 52 mwishopmatrix. 