1. **Problem statement:** Determine the mutual position of the lines \(g\) and \(h\) and find the intersection point if they intersect. Given: \[g: \vec{x} = \begin{pmatrix}1 \\ 5 \\ 2\end{pmatrix} + r \cdot \begin{pmatrix}1 \\ 0 \\ 2\end{pmatrix}, \quad h: \vec{x} = \begin{pmatrix}6 \\ 6 \\ 10\end{pmatrix} + s \cdot \begin{pmatrix}3 \\ 1 \\ 4\end{pmatrix}\] 2. **Formula and rules:** - Two lines intersect if there exist parameters \(r\) and \(s\) such that their position vectors are equal. - Set \(\vec{x}_g = \vec{x}_h\) and solve for \(r\) and \(s\). 3. **Set up the system:** \[ \begin{cases} 1 + r = 6 + 3s \\ 5 + 0 \cdot r = 6 + s \\ 2 + 2r = 10 + 4s \end{cases} \] 4. **Rewrite equations:** \[ \begin{cases} r - 3s = 5 \\ 5 = 6 + s \\ 2 + 2r = 10 + 4s \end{cases} \] 5. **Solve second equation for \(s\):** \[ s = 5 - 6 = -1\] 6. **Substitute \(s = -1\) into first equation:** \[ r - 3(-1) = 5 \implies r + 3 = 5 \implies r = 2\] 7. **Substitute \(r=2\), \(s=-1\) into third equation:** \[ 2 + 2 \cdot 2 = 10 + 4 \cdot (-1) \\ 2 + 4 = 10 - 4 \\ 6 = 6 \quad \checkmark \] 8. **Conclusion:** The parameters satisfy all equations, so the lines intersect. 9. **Find intersection point:** \[ \vec{x} = \begin{pmatrix}1 \\ 5 \\ 2\end{pmatrix} + 2 \cdot \begin{pmatrix}1 \\ 0 \\ 2\end{pmatrix} = \begin{pmatrix}1 + 2 \\ 5 + 0 \\ 2 + 4\end{pmatrix} = \begin{pmatrix}3 \\ 5 \\ 6\end{pmatrix}\] **Final answer:** The lines \(g\) and \(h\) intersect at the point \( (3, 5, 6) \).