Let $P$ be the plane $\sqrt { 3 } x + 2 y + 3 z = 16$ and let $S = \left\{ \alpha \hat { i } + \beta \hat { j } + \gamma \hat { k } : \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 1 \right.$ and the distance of $( \alpha , \beta , \gamma )$ from the plane $P$ is $\left. \frac { 7 } { 2 } \right\}$. Let $\vec { u } , \vec { v }$ and $\vec { w }$ be three distinct vectors in $S$ such that $| \vec { u } - \vec { v } | = | \vec { v } - \vec { w } | = | \vec { w } - \vec { u } |$. Let $V$ be the volume of the parallelepiped determined by vectors $\vec { u } , \vec { v }$ and $\vec { w }$. Then the value of $\frac { 80 } { \sqrt { 3 } } V$ is