Find the volume of the parallelopiped whose sides are given by the vectors :
(i) `11hat(i), 2hat(j), 13 hat(k)`
(ii) `3hat(i)+4hat(j), 2hat(i)+3hat(j)+4hat(k), 5hat(k)`.

Answer

To find the volume of the parallelepiped defined by the given vectors, we can use the scalar triple product, which can be computed using the determinant of a 3x3 matrix formed by the vectors. Let's solve both parts step by step.

### Part (i)

Given vectors:
- \( \mathbf{A} = 11\hat{i} \)
- \( \mathbf{B} = 2\hat{j} \)
- \( \mathbf{C} = 13\hat{k} \)

1. **Set up the determinant**:
The volume \( V \) of the parallelepiped can be calculated using the determinant:
\[ V = \left| \begin{array}{ccc} 11 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 13 \end{array} \right| \]

2. **Calculate the determinant**:
The determinant of a diagonal matrix is the product of its diagonal elements:
\[ V = 11 \times 2 \times 13 = 286 \]

3. **Final result**:
The volume of the parallelepiped is \( 286 \) cubic units.

### Part (ii)

Given vectors:
- \( \mathbf{A} = 3\hat{i} + 4\hat{j} \)
- \( \mathbf{B} = 2\hat{i} + 3\hat{j} + 4\hat{k} \)
- \( \mathbf{C} = 5\hat{k} \)

1. **Set up the determinant**:
The volume \( V \) can be calculated using the determinant:
\[ V = \left| \begin{array}{ccc} 3 & 4 & 0 \\ 2 & 3 & 4 \\ 0 & 0 & 5 \end{array} \right| \]

2. **Calculate the determinant**:
We can expand the determinant using the first row:
\[ V = 3 \left| \begin{array}{cc} 3 & 4 \\ 0 & 5 \end{array} \right| - 4 \left| \begin{array}{cc} 2 & 4 \\ 0 & 5 \end{array} \right| + 0 \]
Calculating the 2x2 determinants:
\[ \left| \begin{array}{cc} 3 & 4 \\ 0 & 5 \end{array} \right| = 3 \cdot 5 - 4 \cdot 0 = 15 \]
\[ \left| \begin{array}{cc} 2 & 4 \\ 0 & 5 \end{array} \right| = 2 \cdot 5 - 4 \cdot 0 = 10 \]
Substituting back:
\[ V = 3 \cdot 15 - 4 \cdot 10 = 45 - 40 = 5 \]

3. **Final result**:
The volume of the parallelepiped is \( 5 \) cubic units.

### Summary of Results
- Volume of the first parallelepiped: \( 286 \) cubic units.
- Volume of the second parallelepiped: \( 5 \) cubic units.