### Instruction: If $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are vectors such that $\mathbf{a} \cdot \mathbf{b} = -3,$ $\mathbf{a} \cdot \mathbf{c} = 4,$ and $\mathbf{b} \cdot \mathbf{c} = 6,$ then find
\[\mathbf{b} \cdot (7 \mathbf{c} - 2 \mathbf{a}).\].### Response: To solve this problem, I will use the fact that the dot product of two vectors is distributive over vector addition and subtraction, and that it is commutative.
That means I can write $\mathbf{b} \cdot (7 \mathbf{c} - 2 \mathbf{a})$ as $7 (\mathbf{b} \cdot \mathbf{c}) - 2 (\mathbf{b} \cdot \mathbf{a}).$
Now I can plug in the given values of the dot products and simplify.