Let $\vec { p } = 2 \hat { i } + \hat { j } + 3 \hat { k }$ and $\vec { q } = \hat { i } - \hat { j } + \hat { k }$. If for some real numbers $\alpha , \beta$, and $\gamma$, we have
$$15 \hat { i } + 10 \hat { j } + 6 \hat { k } = \alpha ( 2 \vec { p } + \vec { q } ) + \beta ( \vec { p } - 2 \vec { q } ) + \gamma ( \vec { p } \times \vec { q } )$$
then the value of $\gamma$ is $\_\_\_\_$ .