question_answer

\[\vec{a},\,\vec{b},\,\vec{c}\] are unit vectors such that \[\left| \vec{a}+\vec{b}+3\vec{c} \right|=\sqrt{14}\] . Angle between \[\vec{a}\] and \[\vec{b}\] is\[\alpha ,\] angle between \[\vec{b}\] and \[\vec{c}\] is \[\beta ,\]and angle between \[\vec{a}\] and \[\vec{c}\] is \[\gamma \]. If \[\gamma \in \left[ \frac{\pi }{6},\frac{2\pi }{3} \right],\]then maximum value of \[\cos \alpha +3\cos \beta \] is ____.

Answer:

3