Step-by-step Solution:

\[ \textbf{Concept:} \] The volume of the parallelepiped determined by \(\vec{a}, \vec{b}\) and \(\vec{c}\) is given by: \[ \text{Volume} = \vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} \] \[ \textbf{Calculation:} \] Parallelepiped determined by: \[ u = i + 2j - k \] \[ v = -2i + 3k \] \[ w = 7j - 4k \] \[ \text{Volume} = \begin{vmatrix} 1 & 2 & -1 \\ -2 & 0 & 3 \\ 0 & 7 & -4 \end{vmatrix} \] \[ \Rightarrow \text{Volume} = |1 \times (0 - 21) - 2 \times (8 - 0) + (-1) \times (-14 - 0)| \] \[ \Rightarrow \text{Volume} = |\boldsymbol{-23}| = 23 \text{ units} \]