Given that \(\mathbf{a} \times \mathbf{( b} \times \mathbf{c})=\frac{\sqrt{3}}{2} \mathbf{b}+\frac{1}{2} \mathbf{c}\)
\(\Rightarrow \quad \text { (a . c) } \mathbf{b}-\mathbf{( a} \cdot \mathbf{b}) \mathbf{c}=\frac{\sqrt{3}}{2} \mathbf{b}+\frac{1}{2} \mathbf{c}\)
Let angle between unit vectors \(\mathbf{a}\) and \(\mathbf{c}\) is ' \(\alpha\) ' and angle between unit vectors \(\mathbf{a}\) and \(\mathbf{b}\) is ' \(\beta\) ', then
\(\begin{array}{rlrl}
& (\cos \alpha) \mathbf{b}-(\cos \beta) \mathbf{c}=\frac{\sqrt{3}}{2} \mathbf{b}+\frac{1}{2} \mathbf{c} \\
\Rightarrow & \cos \alpha=\frac{\sqrt{3}}{2} \text { and } \cos \beta=-\frac{1}{2} \\
\Rightarrow & & \alpha=30^{\circ} \text { and } \beta=120^{\circ}
\end{array}\)
Hence, option (b) is correct.