The area of parallelogram represented by the vectors `overset(rarr)A = 2 hat i + 3 hat j` and `overset(rarr)B=hat i+4 hat j` is

Explanation

To find the area of the parallelogram represented by the vectors \(\overset{\rarr}{A} = 2 \hat{i} + 3 \hat{j}\) and \(\overset{\rarr}{B} = \hat{i} + 4 \hat{j}\), we can use the formula for the area of a parallelogram formed by two vectors, which is given by the magnitude of their cross product.

### Step-by-Step Solution:

1. **Identify the Vectors**:
- Let \(\overset{\rarr}{A} = 2 \hat{i} + 3 \hat{j}\)
- Let \(\overset{\rarr}{B} = \hat{i} + 4 \hat{j}\)

2. **Set Up the Cross Product**:
The cross product \(\overset{\rarr}{A} \times \overset{\rarr}{B}\) can be calculated using the determinant of a matrix formed by the unit vectors \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\) and the components of the vectors \(\overset{\rarr}{A}\) and \(\overset{\rarr}{B}\).

\[ \overset{\rarr}{A} \times \overset{\rarr}{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & 0 \\ 1 & 4 & 0 \end{vmatrix} \]

3. **Calculate the Determinant**:
- Expand the determinant:
\[ \overset{\rarr}{A} \times \overset{\rarr}{B} = \hat{i} \begin{vmatrix} 3 & 0 \\ 4 & 0 \end{vmatrix} - \hat{j} \begin{vmatrix} 2 & 0 \\ 1 & 0 \end{vmatrix} + \hat{k} \begin{vmatrix} 2 & 3 \\ 1 & 4 \end{vmatrix} \]

- Calculate each of the 2x2 determinants:
- For \(\hat{i}\):
\[ 3 \cdot 0 - 4 \cdot 0 = 0 \]
- For \(\hat{j}\):
\[ 2 \cdot 0 - 1 \cdot 0 = 0 \]
- For \(\hat{k}\):
\[ 2 \cdot 4 - 3 \cdot 1 = 8 - 3 = 5 \]

4. **Combine the Results**:
- So, we have:
\[ \overset{\rarr}{A} \times \overset{\rarr}{B} = 0 \hat{i} - 0 \hat{j} + 5 \hat{k} = 5 \hat{k} \]

5. **Find the Magnitude**:
- The magnitude of the cross product gives the area of the parallelogram:
\[ \text{Area} = |\overset{\rarr}{A} \times \overset{\rarr}{B}| = |5 \hat{k}| = 5 \]

### Final Answer:
The area of the parallelogram is \(5\).