The problem asks to find the value of $k$ such that the vectors $\vec{a} = \langle 2, 3, 1 \rangle$, $\vec{b} = \langle 1, 2, 3 \rangle$, and $\vec{c} = \langle 18, 5, k \rangle$ all lie on the same plane.

To check if three vectors lie on the same plane, we can check if the scalar triple product of these vectors equals zero. The scalar triple product of vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ is given by the determinant: