<prompt> Kuna vekta mathbfa na mathbfb vilemathbfa mathbfb anzapmatrix 6 -3 -6 mwishopmatrix,ambapo mathbfa ni sambamba na begin pmatrix 1 1 1 mwishopmatrix, na mathbfb ni orthogonal kwa beginpmatrix 1 1 1 endpmatrix. Find mathbfb.  <response>: Kwa kuwa mathbfa ni sambamba na beginpmatrix 1 1 1 endpmatrix,mathbfa t anzapmatrix 1 1 1 mwishopmatrix anzapmatrix t t t mwishopmatrixkwa baadhi ya scalar t. Kishamathbfb anzapmatrix 6 -3 -6 mwishopmatrix - anzapmatrix t t t mwishopmatrix anzapmatrix 6 - t -3 - t -6 - t mwishopmatrix.Tunataka hii iwe orthogonal kwa anzapmatrix 1 1 1 mwishopmatrix, hivyoanzapmatrix 6 - t -3 - t -6 - t mwishopmatrix cdot anzapmatrix 1 1 1 mwishopmatrix 0. Kisha 6 - t1 -3 - t1 -6 - t1 0. Kutatua, tunapata t -1. Kisha mathbf b anzapmatrix 7 -2 -5 mwishopmatrix. 