The diagonals of a parallelogram are given by the vectors `(3 hat(i) + hat(j) + 2hat(k)) and ( hat(i) - 3hat(j) + 4hat(k))` in m . Find the area of the parallelogram .

Explanation

To find the area of the parallelogram given the diagonals represented by the vectors \( \mathbf{d_1} = 3\hat{i} + \hat{j} + 2\hat{k} \) and \( \mathbf{d_2} = \hat{i} - 3\hat{j} + 4\hat{k} \), we can follow these steps:

### Step 1: Identify the vectors
The diagonals of the parallelogram are given as:
\[ \mathbf{d_1} = 3\hat{i} + \hat{j} + 2\hat{k} \]
\[ \mathbf{d_2} = \hat{i} - 3\hat{j} + 4\hat{k} \]

### Step 2: Calculate the cross product \( \mathbf{d_1} \times \mathbf{d_2} \)
To find the area of the parallelogram, we need to compute the cross product of the two diagonal vectors. The formula for the cross product in determinant form is:
\[ \mathbf{d_1} \times \mathbf{d_2} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 1 & 2 \\ 1 & -3 & 4 \end{vmatrix} \]

### Step 3: Calculate the determinant
Calculating the determinant, we expand it as follows:
\[ \mathbf{d_1} \times \mathbf{d_2} = \hat{i} \begin{vmatrix} 1 & 2 \\ -3 & 4 \end{vmatrix} - \hat{j} \begin{vmatrix} 3 & 2 \\ 1 & 4 \end{vmatrix} + \hat{k} \begin{vmatrix} 3 & 1 \\ 1 & -3 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
1. For \( \hat{i} \):
\[ \begin{vmatrix} 1 & 2 \\ -3 & 4 \end{vmatrix} = (1)(4) - (2)(-3) = 4 + 6 = 10 \]

2. For \( \hat{j} \):
\[ \begin{vmatrix} 3 & 2 \\ 1 & 4 \end{vmatrix} = (3)(4) - (2)(1) = 12 - 2 = 10 \]

3. For \( \hat{k} \):
\[ \begin{vmatrix} 3 & 1 \\ 1 & -3 \end{vmatrix} = (3)(-3) - (1)(1) = -9 - 1 = -10 \]

Putting it all together:
\[ \mathbf{d_1} \times \mathbf{d_2} = 10\hat{i} - 10\hat{j} - 10\hat{k} \]

### Step 4: Find the magnitude of the cross product
Now, we find the magnitude of the vector \( \mathbf{d_1} \times \mathbf{d_2} \):
\[ |\mathbf{d_1} \times \mathbf{d_2}| = \sqrt{(10)^2 + (-10)^2 + (-10)^2} = \sqrt{100 + 100 + 100} = \sqrt{300} = 10\sqrt{3} \]

### Step 5: Calculate the area of the parallelogram
The area \( A \) of the parallelogram is given by:
\[ A = \frac{1}{2} |\mathbf{d_1} \times \mathbf{d_2}| = \frac{1}{2} (10\sqrt{3}) = 5\sqrt{3} \]

### Step 6: Approximate the area
Using \( \sqrt{3} \approx 1.73 \):
\[ A \approx 5 \times 1.73 \approx 8.65 \, \text{m}^2 \]

Thus, the area of the parallelogram is approximately \( 8.65 \, \text{m}^2 \).