If $$\vec{a} = \hat{i} + \hat{j} + \hat{k}$$, $$\vec{b} = \hat{j} - \hat{k}$$ and $$\vec{c}$$ be three vectors such that $$\vec{a} \times \vec{c} = \vec{b}$$ and $$\vec{a} \cdot \vec{c} = 3$$, then $$\vec{c} \cdot (\vec{a} - 2\vec{b})$$ is equal to _______.

$$\text{Expression} = \vec{c} \cdot \vec{a} - 2(\vec{c} \cdot \vec{b})$$

$$\text{Expression} = 3 - 2(\vec{c} \cdot \vec{b}) \quad \text{--- (Equation 1)}$$

$$\vec{b} = \vec{a} \times \vec{c}$$

$$\vec{c} \cdot \vec{b} = \vec{c} \cdot (\vec{a} \times \vec{c})$$

$$\vec{c} \cdot (\vec{a} \times \vec{c}) = 0$$

$$ \implies$$ $$\vec{c} \cdot \vec{b} = 0$$

$$\text{Expression} = 3 - 2(0) = 3$$