Dot Product of Vector Difference with Given Vectors

To evaluate the expression $\vec{c} \cdot (\vec{a} - 2\vec{b})$, we can expand it using the distributive property of the dot product:

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

From the problem statement, we are given:

1. $\vec{a} \cdot \vec{c} = 3$ (so, $\vec{c} \cdot \vec{a} = 3$)
2. $\vec{a} \times \vec{c} = \vec{b}$

Since the cross product of two vectors is orthogonal to both vectors involved, the vector $\vec{b} = \vec{a} \times \vec{c}$ is perpendicular to $\vec{c}$.

Therefore, the dot product of $\vec{c}$ and $\vec{b}$ must be zero: $\vec{c} \cdot \vec{b} = 0$

Substituting these values back into our expanded expression:

$\vec{c} \cdot (\vec{a} - 2\vec{b}) = 3 - 2(0) = 3$