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

Explanation

To find the volume of the parallelepiped formed 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 will use the formula for the volume of a parallelepiped given by the scalar triple product of the vectors. The volume \(V\) can be calculated as:

\[ V = |\vec{a} \cdot (\vec{b} \times \vec{c})| \]

### Step 1: Calculate the cross product \(\vec{b} \times \vec{c}\)

To find \(\vec{b} \times \vec{c}\), we can use the determinant of a matrix formed by the unit vectors and the components of \(\vec{b}\) and \(\vec{c}\):

\[ \vec{b} \times \vec{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & -1 \\ 3 & -1 & 2 \end{vmatrix} \]

Calculating this determinant:

\[ = \hat{i} \begin{vmatrix} 2 & -1 \\ -1 & 2 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & -1 \\ 3 & 2 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 2 \\ 3 & -1 \end{vmatrix} \]

Calculating the 2x2 determinants:

1. \(\begin{vmatrix} 2 & -1 \\ -1 & 2 \end{vmatrix} = (2)(2) - (-1)(-1) = 4 - 1 = 3\)
2. \(\begin{vmatrix} 1 & -1 \\ 3 & 2 \end{vmatrix} = (1)(2) - (-1)(3) = 2 + 3 = 5\)
3. \(\begin{vmatrix} 1 & 2 \\ 3 & -1 \end{vmatrix} = (1)(-1) - (2)(3) = -1 - 6 = -7\)

Putting it all together:

\[ \vec{b} \times \vec{c} = 3\hat{i} - 5\hat{j} - 7\hat{k} \]

### Step 2: Calculate the dot product \(\vec{a} \cdot (\vec{b} \times \vec{c})\)

Now we need to calculate the dot product:

\[ \vec{a} \cdot (\vec{b} \times \vec{c}) = (2\hat{i} - 3\hat{j} + 4\hat{k}) \cdot (3\hat{i} - 5\hat{j} - 7\hat{k}) \]

Calculating the dot product:

\[ = 2 \cdot 3 + (-3) \cdot (-5) + 4 \cdot (-7) \]
\[ = 6 + 15 - 28 \]
\[ = 21 - 28 = -7 \]

### Step 3: Calculate the volume

Finally, the volume \(V\) is the absolute value of the dot product:

\[ V = |\vec{a} \cdot (\vec{b} \times \vec{c})| = |-7| = 7 \]

Thus, the volume of the parallelepiped is \(7\) cubic units.

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