Let $\vec { x }$ be a vector in the plane containing vectors $\vec { a } = 2 \hat { i } - \hat { j } + \hat { k }$ and $\vec { b } = \hat { i } + 2 \hat { j } - \hat { k }$. If the vector $\vec { x }$ is perpendicular to $( 3 \hat { i } + 2 \hat { j } - \widehat { k } )$ and its projection on $\vec { a }$ is $\frac { 17 \sqrt { 6 } } { 2 }$, then the value of $| \vec { x } | ^ { 2 }$ is equal to $\_\_\_\_$ .