<prompt> Acha mathbfu na mathbfv ziwe viveta vya kitengo, na mathbfw iwe vekta ilimathbfw mathbfw nyakati mathbfu mathbfv.Tafuta thamani kubwa iwezekanavyo ya mathbfu times mathbfv cdot mathbfw.  <response>: Kutoka mathbfw mathbfw nyakati mathbfu mathbfv,mathbfw nyakati mathbfu mathbfv - mathbfw.Kishaanzaalignmathbfw nyakati mathbfu2 mathbfv - mathbfw2 mathbfv2 - 2 mathbfv cdot mathbfw mathbfw2 1 - 2 mathbfv cdot mathbfw mathbfw2.mwishoalignKwa hiyo,mathbfv cdot mathbfw frac1 mathbfw2 - mathbfw nyakati mathbfu2 2. quad Pia kutoka mathbfw mathbfw times mathbfu mathbfv, tunaweza kuchukua bidhaa ya nukta na mathbfv, kupatamathbfw cdot mathbfv mathbfw nyakati mathbfu cdot mathbfv mathbfv cdot mathbfv 1.Kwa bidhaa ya scalar triple, mathbfw times mathbfu cdot mathbfv mathbfu mara mathbfv cdot mathbfw, hivyomathbfu nyakati mathbfv cdot mathbfw 1 - mathbfv cdot mathbfw.Kutoka mlinganyo ,anzaalignmathbfu nyakati mathbfv cdot mathbfw 1 - frac1 mathbfw2 - mathbfw nyakati mathbfu22 frac12 - frac12 mathbfw2 frac12 mathbfw mara mathbf u2.endalignAcha theta iwe pembe kati ya mathbfu na mathbfw. Kishaanzaalignmathbfu nyakati mathbfv cdot mathbfw frac12 - frac12 mathbfw 2 frac12 mathbfw nyakati mathbfu2 frac12 - frac12 mathbfw2 frac12 mathbfu2 mathbfw2 sin2 theta frac12 - frac12 mathbfw2 frac12 mathbfw2 sin2 theta frac12 - frac12 mathbfw2 cos2 theta le frac12.endalignUsawa hutokea wakati mathbfu beginpmatrix 1 0 0 endpmatrix, mathbfv beginpmatrix 0 1 0 endpmatrix, na mathbfw anzapmatrix 0 12 12 endpmatrix, hivyo thamani kubwa iwezekanavyo ya mathbfu nyakati mathbfv cdot mathbfw ni frac12. 