<prompt> Acha mathbfa anzepmatrix 1 3 0 mwishopmatrix, mathbfb anzapmatrix -11 5 2 mwisho pmatrix, na mathbfc anzapmatrix 1 sqrt5 4 -5 endpmatrix. Tafuta k ikiwa vekta mathbfa mathbfb mathbfc na3 mathbfb nyakati mathbfc - 8 mathbfc nyakati mathbfa k mathbfa nyakati mathbfb zina othogonal.  <response>: Tangu mathbfa mathbfb mathbfc na 3 mathbfb mara mathbfc - 8 mathbfc nyakati mathbfa k mathbfa nyakati mathbfb ni za othogonal,mathbfa mathbfb mathbfc cdot 3 mathbfb nyakati mathbfc - 8 mathbfc nyakati mathbfa k mathbfa nyakati mathbfb 0.Kupanua, tunapataanzaalign3 mathbfa cdot mathbfb mara mathbfc - 8 mathbfa cdot mathbfc nyakati mathbfa k mathbfa cdot mathbfa nyakati mathbfb quad 3 mathbfb cdot mathbfb nyakati mathbfc - 8 mathbfb cdot mathbfc nyakati mathbfa k mathbfb cdot mathbfa nyakati mathbfb quad 3 mathbfc cdot mathbfb nyakati mathbfc - 8 mathbfc cdot mathbfc nyakati mathbfa k mathbfc cdot mathbfa nyakati mathbfb 0.endalignKwa vile mathbfa na mathbfc nyakati mathbfa ni za orthogonal, bidhaa zao za nukta ni 0. Vilevile, maneno mengi hutoweka, nasi tumeachwa. na3 mathbfa cdot mathbfb mara mathbfc - 8 mathbfb cdot mathbfc nyakati mathbfa k mathbfc cdot mathbfa times mathbfb 0.Kwa bidhaa ya scalar triple,mathbfa cdot mathbfb nyakati mathbfc mathbfb cdot mathbfc nyakati mathbfa mathbfc cdot mathbfa nyakati mathbfb,hivyo 3 - 8 k mathbfa cdot mathbfb nyakati mathbfc 0. Tunaweza kuthibitisha kwamba mathbfa cdot mathbfb mara mathbfc neq 0, kwa hivyo ni lazima tuwe na 3 - 8 k 0, ambayo ina maana k 5. 