Find the volume of the parallelopiped whose edges are represented by `vec(a) = 2 hat(i) - 3 hat(j) + 4 hat(k) , vec(b) = hat(i) + 2 hat(j) - hat(k), vec(c) = 3 hat(i) - hat(j) + 2 hat(k)`

Answer

To find the volume of the parallelepiped whose edges are represented by the vectors \(\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}\), \(\vec{b} = \hat{i} + 2\hat{j} - \hat{k}\), and \(\vec{c} = 3\hat{i} - \hat{j} + 2\hat{k}\), we can use the formula for the volume of a parallelepiped formed by three vectors, which is given by the absolute value of the scalar triple product of the vectors. This can be calculated using the determinant of a 3x3 matrix formed by the components of the vectors.

### Step-by-Step Solution:

1. **Write the vectors in component form:**
\[ \vec{a} = \begin{pmatrix} 2 \\ -3 \\ 4 \end{pmatrix}, \quad \vec{b} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}, \quad \vec{c} = \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix} \]

2. **Set up the determinant:**
The volume \(V\) of the parallelepiped can be calculated as:
\[ V = |\vec{a} \cdot (\vec{b} \times \vec{c})| = \left| \begin{vmatrix} 2 & -3 & 4 \\ 1 & 2 & -1 \\ 3 & -1 & 2 \end{vmatrix} \right| \]

3. **Calculate the determinant:**
We can calculate the determinant using the formula for a 3x3 matrix:
\[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg) \]
Applying this to our matrix:
\[ = 2 \begin{vmatrix} 2 & -1 \\ -1 & 2 \end{vmatrix} - (-3) \begin{vmatrix} 1 & -1 \\ 3 & 2 \end{vmatrix} + 4 \begin{vmatrix} 1 & 2 \\ 3 & -1 \end{vmatrix} \]

4. **Calculate the 2x2 determinants:**
- For the first determinant:
\[ \begin{vmatrix} 2 & -1 \\ -1 & 2 \end{vmatrix} = (2)(2) - (-1)(-1) = 4 - 1 = 3 \]
- For the second determinant:
\[ \begin{vmatrix} 1 & -1 \\ 3 & 2 \end{vmatrix} = (1)(2) - (-1)(3) = 2 + 3 = 5 \]
- For the third determinant:
\[ \begin{vmatrix} 1 & 2 \\ 3 & -1 \end{vmatrix} = (1)(-1) - (2)(3) = -1 - 6 = -7 \]

5. **Substitute back into the determinant calculation:**
\[ = 2(3) + 3(5) + 4(-7) = 6 + 15 - 28 = 21 - 28 = -7 \]

6. **Find the volume:**
The volume is the absolute value of the determinant:
\[ V = |-7| = 7 \]

### Final Answer:
The volume of the parallelepiped is \(7\) cubic units.