Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors. Let $$|\vec{a}| = 1$$, $$|\vec{b}| = 4$$ and $$\vec{a} \cdot \vec{b} = 2$$. If $$\vec{c} = (2\vec{a} \times \vec{b}) - 3\vec{b}$$, then the value of $$\vec{b} \cdot \vec{c}$$ is

Given $$|\vec{a}| = 1$$, $$|\vec{b}| = 4$$, $$\vec{a} \cdot \vec{b} = 2$$, and $$\vec{c} = (2\vec{a} \times \vec{b}) - 3\vec{b}$$. Find $$\vec{b} \cdot \vec{c}$$.

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

$$\vec{b} \cdot (\vec{a} \times \vec{b})$$ is the scalar triple product $$[\vec{b}, \vec{a}, \vec{b}]$$. Since two vectors in the scalar triple product are identical, this equals zero.

$$\vec{b} \cdot (2\vec{a} \times \vec{b}) = 2 \times 0 = 0$$

$$\vec{b} \cdot \vec{b} = |\vec{b}|^2 = 16$$

$$\vec{b} \cdot \vec{c} = 0 - 3(16) = -48$$

The correct answer is Option B: $$-48$$.