Consider the vectors $$\vec { x } = \hat { \imath } + 2 \hat { \jmath } + 3 \hat { k } , \quad \vec { y } = 2 \hat { \imath } + 3 \hat { \jmath } + \hat { k } , \quad \text { and } \quad \vec { z } = 3 \hat { \imath } + \hat { \jmath } + 2 \hat { k }$$ For two distinct positive real numbers $\alpha$ and $\beta$, define $$\vec { X } = \alpha \vec { x } + \beta \vec { y } - \vec { z } , \quad \vec { Y } = \alpha \vec { y } + \beta \vec { z } - \vec { x } , \quad \text { and } \quad \vec { Z } = \alpha \vec { z } + \beta \vec { x } - \vec { y }$$ If the vectors $\vec { X } , \vec { Y }$, and $\vec { Z }$ lie in a plane, then the value of $\alpha + \beta - 3$ is $\_\_\_\_$.
Consider the vectors
$$\vec { x } = \hat { \imath } + 2 \hat { \jmath } + 3 \hat { k } , \quad \vec { y } = 2 \hat { \imath } + 3 \hat { \jmath } + \hat { k } , \quad \text { and } \quad \vec { z } = 3 \hat { \imath } + \hat { \jmath } + 2 \hat { k }$$
For two distinct positive real numbers $\alpha$ and $\beta$, define
$$\vec { X } = \alpha \vec { x } + \beta \vec { y } - \vec { z } , \quad \vec { Y } = \alpha \vec { y } + \beta \vec { z } - \vec { x } , \quad \text { and } \quad \vec { Z } = \alpha \vec { z } + \beta \vec { x } - \vec { y }$$
If the vectors $\vec { X } , \vec { Y }$, and $\vec { Z }$ lie in a plane, then the value of $\alpha + \beta - 3$ is $\_\_\_\_$.