To solve the problem, we start with the given equation:

$$(\vec{a} - \vec{c}) \times \vec{b} = -18 \hat{i} - 3 \hat{j} + 12 \hat{k}.$$

Expanding this, we have:

$$\vec{a} \times \vec{b} - \vec{c} \times \vec{b} = -18 \hat{i} - 3 \hat{j} + 12 \hat{k}.$$

Let $\vec{d} = \vec{c} \times \vec{b}$. Substituting this into the equation, we get:

$$\vec{a} \times \vec{b} + \vec{d} = -18 \hat{i} - 3 \hat{j} + 12 \hat{k}.$$

Now, we take the dot product of both sides with $\vec{a}$:

$$\vec{a} \cdot (\vec{a} \times \vec{b}) + \vec{a} \cdot \vec{d} = \vec{a} \cdot (-18 \hat{i} - 3 \hat{j} + 12 \hat{k}).$$

Since the dot product $\vec{a} \cdot (\vec{a} \times \vec{b})$ is zero (the dot product of any vector with the cross product of itself with another vector is zero), we have:

$$\vec{a} \cdot \vec{d} = -36 + 9 + 12 = -15.$$

Thus, the magnitude of $\vec{a} \cdot \vec{d}$ is:

$$|\vec{a} \cdot \vec{d}| = 15.$$