To find the volume of the parallelepiped defined by the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), we need to calculate the scalar triple product of these vectors. The volume \(V\) is given by:
\(V = |\vec{a} \cdot (\vec{b} \times \vec{c})|\)
First, let's compute \(\vec{b} \times \vec{c}\):
Using the formula for the cross product of two vectors:
\(\vec{b} \times \vec{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -1 & 2 \\ 2 & 1 & -1 \end{vmatrix}\)
Calculating the determinant:
\(\vec{b} \times \vec{c} = \hat{i}((-1) \cdot (-1) - 2 \cdot 1) - \hat{j}(1 \cdot (-1) - 2 \cdot 2) + \hat{k}(1 \cdot 1 - (-1) \cdot 2)\)
\(= \hat{i}(1 - 2) - \hat{j}(-1 - 4) + \hat{k}(1 + 2)\)
\(= -\hat{i} + 5\hat{j} + 3\hat{k}\)
Now, compute \(\vec{a} \cdot (\vec{b} \times \vec{c})\):
\(\vec{a} \cdot (-\hat{i} + 5\hat{j} + 3\hat{k}) = (2\hat{i} - 3\hat{j} + \hat{k}) \cdot (-\hat{i} + 5\hat{j} + 3\hat{k})\)
\(= 2 \cdot (-1) + (-3) \cdot 5 + 1 \cdot 3\)
\(= -2 - 15 + 3\)
\(= -14\)
The volume of the parallelepiped is the absolute value of this result:
\(V = |-14| = 14\)
Therefore, the volume of the parallelepiped is 14.