Solution: Let $ \theta $ be the angle between $ \mathbfa $ and $ \mathbfb $, so $ \cos\theta = \frac12 \Rightarrow \theta = 60^\circ $. Let $ \phi $ be the angle between $ \mathbfb $ and $ \mathbfc $, so $ \cos\phi = \frac\sqrt32 \Rightarrow \phi = 30^\circ $. To maximize $ \mathbfa \cdot \mathbfc = \cos(\alpha) $, where $ \alpha $ is the angle between $ \mathbfa $ and $ \mathbfc $, arrange $ \mathbfa, \mathbfb, \mathbfc $ in a plane. The maximum occurs when $ \mathbfa $ and $ \mathbfc $ are aligned, but constrained by their angles relative to $ \mathbfb $. The minimum angle between $ \mathbfa $ and $ \mathbfc $ is $ 60^\circ - 30^\circ = 30^\circ $, so $ \cos(30^\circ) = \frac\sqrt32 $. However, if they are aligned, $ \alpha = 0^\circ $, but this requires $ \theta = \phi = 0^\circ $, which contradicts the given dot products. Instead, use the cosine law for angles: $ \cos\alpha \leq \cos(60^\circ - 30^\circ) = \cos(30^\circ) = \frac\sqrt32 $. Thus, the maximum is $ \boxed\frac\sqrt32 $. - Appfinity Technologies