Let $\mathbf{v}$ and $\mathbf{w}$ be vectors such that $\|\mathbf{v}\| = 3$ and $\|\mathbf{w}\| = 4.$  Find the minimum value of $\mathbf{v} \cdot \mathbf{w}.$
I know that the dot product of two vectors can be written in terms of their magnitudes and the angle between them, as $\mathbf{v} \cdot \mathbf{w} = \|\mathbf{v}\|\|\mathbf{w}\|\cos \theta.$ So, to minimize this expression, I want to minimize the cosine of the angle, which means maximizing the angle itself. The largest possible angle between two vectors is 180 degrees, or $\pi$ radians, and the cosine of that is -1. So, the minimum value of $\mathbf{v} \cdot \mathbf{w}$ is obtained when the vectors are opposite in direction, and it is $- \|\mathbf{v}\|\|\mathbf{w}\| = -3 \cdot 4 = -12.$ 