<prompt> Acha mathbfu na mathbfv ziwe viveta vya kitengo, na mathbfw iwe vekta kiasi kwamba mathbfu times mathbfv mathbf u mathbfw na mathbfw nyakati mathbfu mathbfv. Kokotoa mathbfu cdot mathbfv nyakati mathbfw.  <response>: Kutoka mathbfu nyakati mathbfv mathbfu mathbfw na mathbfw nyakati mathbfu mathbfv,mathbfu nyakati mathbfv mathbfu nyakati mathbfu mathbfv.Kupanua, tunapatamathbfu nyakati mathbfv nyakati mathbfu mathbfu nyakati mathbfu mathbfv.Tunajua kwamba mathbfu nyakati mathbfu mathbf0. Kwa bidhaa ya vekta tatu, kwa vekta yoyote mathbfp, mathbfq, na mathbfr ,mathbfp nyakati mathbfq mara mathbfr mathbfp cdot mathbfr mathbfq - mathbfp cdot mathbfq mathbfr.Kwa hivyo,mathbfu cdot mathbfu mathbfv - mathbfu cdot mathbfv mathbfu mathbfv. Tangu mathbfu 1, mathbfv - mathbfu cdot mathbfv mathbfu mathbfv. Kishamathbfu cdot mathbfv mathbfu mathbf0.Tena, tangu mathbfu 1, lazima tuwe na mathbfu cdot mathbfv 0.Sasa,anzaalignmathbfu cdot mathbfv nyakati mathbfw mathbfu cdot mathbfv nyakati mathbfu nyakati mathbfv mathbfu mathbfu cdot mathbfv nyakati mathbfu nyakati mathbfv mathbfv nyakati mathbfu mathbfu cdot mathbfv nyakati mathbfu nyakati mathbfv mathbfu cdot mathbfv nyakati mathbf u.endalignKwa vekta triple product,mathbfv nyakati mathbfu nyakati mathbfv mathbfv cdot mathbfv mathbfu - mathbfv cdot mathbfu mathbfu.Tangu mathbfv 1 na mathbfu cdot mathbfv 0, hii hurahisisha mathbfu. Pia, mathbfu ni orthogonal kwa mathbfv mara mathbfu, hivyomathbfu cdot mathbfv mara mathbfw mathbfu cdot mathbfu 1. 