Let \(\mathbf{a},\) \(\mathbf{b},\) and \(\mathbf{c}\) be unit vectors such that \(\mathbf{a} \cdot \mathbf{b} = \cos \theta\) and \(\mathbf{a} \cdot \mathbf{c} = \sin \theta\). The line \(L_1\) is described by the parametric equation 
\[\mathbf{r}_1(t) = \mathbf{a} + t (\mathbf{a} \times \mathbf{b}),\]
and the line \(L_2\) is described by 
\[\mathbf{r}_2(u) = \mathbf{b} + u (\mathbf{c} \times \mathbf{a}).\]
Determine the value of \(\theta\) if the lines \(L_1\) and \(L_2\) intersect. Let's think step by step and output the final answer within \boxed{}.
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