The adjacent sides of a parallelogram is represented by vectors `2 hat(i) + 3 hat(j) and hat(i) + 4hat(j)`. The area of the parallelogram is

Options

A. 5 units
B. 3 units
C. 8 units
D. 11 units

Answer

To find the area of the parallelogram formed by the vectors \( \mathbf{a} = 2 \hat{i} + 3 \hat{j} \) and \( \mathbf{b} = \hat{i} + 4 \hat{j} \), we can use the formula for the area of a parallelogram defined 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**:
Let
\[ \mathbf{a} = 2 \hat{i} + 3 \hat{j} \]
\[ \mathbf{b} = \hat{i} + 4 \hat{j} \]

2. **Set Up the Cross Product**:
The area of the parallelogram can be found using the cross product \( \mathbf{a} \times \mathbf{b} \). We will use the determinant method to calculate this.

The cross product can be represented as:
\[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & 0 \\ 1 & 4 & 0 \end{vmatrix} \]

3. **Calculate the Determinant**:
To evaluate the determinant, we can expand it as follows:
\[ \mathbf{a} \times \mathbf{b} = \hat{i} \begin{vmatrix} 3 & 0 \\ 4 & 0 \end{vmatrix} - \hat{j} \begin{vmatrix} 2 & 0 \\ 1 & 0 \end{vmatrix} + \hat{k} \begin{vmatrix} 2 & 3 \\ 1 & 4 \end{vmatrix} \]

Now, calculating each of these 2x2 determinants:
- For \( \hat{i} \):
\[ \begin{vmatrix} 3 & 0 \\ 4 & 0 \end{vmatrix} = (3 \cdot 0) - (0 \cdot 4) = 0 \]
- For \( \hat{j} \):
\[ \begin{vmatrix} 2 & 0 \\ 1 & 0 \end{vmatrix} = (2 \cdot 0) - (0 \cdot 1) = 0 \]
- For \( \hat{k} \):
\[ \begin{vmatrix} 2 & 3 \\ 1 & 4 \end{vmatrix} = (2 \cdot 4) - (3 \cdot 1) = 8 - 3 = 5 \]

Putting it all together:
\[ \mathbf{a} \times \mathbf{b} = 0 \hat{i} - 0 \hat{j} + 5 \hat{k} = 5 \hat{k} \]

4. **Find the Magnitude of the Cross Product**:
The magnitude of the cross product \( |\mathbf{a} \times \mathbf{b}| \) is:
\[ |\mathbf{a} \times \mathbf{b}| = |5 \hat{k}| = 5 \]

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