To determine the value of $ x $, given the vectors $\vec{A}=(-x \hat{i} - 6 \hat{j} - 2 \hat{k})$, $\vec{B}=(-\hat{i} + 4 \hat{j} + 3 \hat{k})$, and $\vec{C}=(-8 \hat{i} - \hat{j} + 3 \hat{k})$, and the condition $\vec{A} \cdot (\vec{B} \times \vec{C}) = 0$, we proceed as follows:

First, we calculate the cross product $\vec{B} \times \vec{C}$:

$ \vec{B} \times \vec{C} = 15 \hat{i} - 21 \hat{j} + 33 \hat{k} $

Next, using the condition $\vec{A} \cdot (\vec{B} \times \vec{C}) = 0$, we compute the dot product:

$ \vec{A} \cdot (\vec{B} \times \vec{C}) = (-x \hat{i} - 6 \hat{j} - 2 \hat{k}) \cdot (15 \hat{i} - 21 \hat{j} + 33 \hat{k}) $

$ \Rightarrow (-x)(15) + (-6)(-21) + (-2)(33) = 0 $

$ \Rightarrow -15x + 126 - 66 = 0 $

Solving this equation for $ x $:

$ -15x + 60 = 0 $

$ \Rightarrow x = 4 $

Thus, the value of $ x $ is $ 4 $.