12. Determine the value of $m$ and $n$ so that the vector $\bar{a}=[2,4,6]$ is perpendicular to both vectors $\bar{b}=(m, n, 10)$ and $\bar{c}=(6, m, n)$ [4 marks]

Step 1 ：Given that the vector $\bar{a}=[2,4,6]$ is perpendicular to both vectors $\bar{b}=(m, n, 10)$ and $\bar{c}=(6, m, n)$, we need to find the values of $m$ and $n$.

Step 2 ：Since the dot product of two perpendicular vectors is zero, we have the following equations:

Step 3 ：$\bar{a} \cdot \bar{b} = 2m + 4n + 60 = 0$

Step 4 ：$\bar{a} \cdot \bar{c} = 4m + 6n + 12 = 0$

Step 5 ：Solving the system of equations, we get $m = \boxed{78}$ and $n = \boxed{-54}$