Let \(\mathbf{u} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\), \(\mathbf{v} = \begin{pmatrix} 4 \\ -1 \\ 2 \end{pmatrix}\), and \(\mathbf{w} = \begin{pmatrix} -2 \\ 3 \\ 1 \end{pmatrix}\). The columns of a matrix are \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\). Find the determinant of the matrix formed by these vectors, and verify if the vectors are orthogonal to each other using their dot products. Let's think step by step and output the final answer within \boxed{}.
user