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

### Step-by-step Solution:

1. **Identify the vectors**:
\[ \vec{A} = 2\hat{i} + 3\hat{j}, \quad \vec{B} = \hat{i} + 4\hat{j} \]

2. **Set up the cross product**:
The cross product \(\vec{A} \times \vec{B}\) in two dimensions can be calculated using the determinant of a matrix formed by the unit vectors and the components of the vectors:
\[ \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & 0 \\ 1 & 4 & 0 \end{vmatrix} \]

3. **Calculate the determinant**:
The determinant can be expanded as follows:
\[ \vec{A} \times \vec{B} = \hat{k} \left( 2 \cdot 4 - 3 \cdot 1 \right) \]
\[ = \hat{k} (8 - 3) = \hat{k} (5) \]

4. **Find the magnitude of the cross product**:
The magnitude of the vector \(\vec{A} \times \vec{B}\) is:
\[ |\vec{A} \times \vec{B}| = 5 \]

5. **Conclusion**:
The area of the parallelogram is equal to the magnitude of the cross product:
\[ \text{Area} = |\vec{A} \times \vec{B}| = 5 \text{ square units} \]

### Final Answer:
The area of the parallelogram represented by the vectors \(\vec{A}\) and \(\vec{B}\) is \(5\) square units.