Given vectors \(\mathbf{u} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}\), \(\mathbf{v} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}\), and \(\mathbf{w} = \begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}\), find the determinant of the matrix \(\mathbf{A}\) whose columns are these vectors. Additionally, verify if \(\mathbf{u}\) is a unit vector and calculate the dot product \(\mathbf{u} \cdot \mathbf{v}\). Let's think step by step and output the final answer within \boxed{}.
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