Two adjacent sides of a parallelogram are respectively by the two vectors `hat(i)+2hat(j)+3hat(k)` and `3hat(i)-2hat(j)+hat(k)`. What is the area of parallelogram?

Options

A. 8
B. `8sqrt3`
C. `3sqrt8`
D. 192

Explanation

To find the area of the parallelogram formed by the two vectors, we will follow these steps:

### Step 1: Identify the Vectors
The two adjacent sides of the parallelogram are given by the vectors:
- Vector A: \(\hat{i} + 2\hat{j} + 3\hat{k}\)
- Vector B: \(3\hat{i} - 2\hat{j} + \hat{k}\)

### Step 2: Set Up the Cross Product
The area of the parallelogram can be calculated using the magnitude of the cross product of the two vectors. The cross product \( \vec{A} \times \vec{B} \) can be computed using the determinant of a matrix formed by the unit vectors and the components of the vectors.

\[ \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 3 \\ 3 & -2 & 1 \end{vmatrix} \]

### Step 3: Calculate the Determinant
To calculate the determinant, we can expand it as follows:

\[ \vec{A} \times \vec{B} = \hat{i} \begin{vmatrix} 2 & 3 \\ -2 & 1 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 3 \\ 3 & 1 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 2 \\ 3 & -2 \end{vmatrix} \]

Calculating each of these 2x2 determinants:

1. For \(\hat{i}\):
\[ \begin{vmatrix} 2 & 3 \\ -2 & 1 \end{vmatrix} = (2)(1) - (3)(-2) = 2 + 6 = 8 \]

2. For \(\hat{j}\):
\[ \begin{vmatrix} 1 & 3 \\ 3 & 1 \end{vmatrix} = (1)(1) - (3)(3) = 1 - 9 = -8 \]

3. For \(\hat{k}\):
\[ \begin{vmatrix} 1 & 2 \\ 3 & -2 \end{vmatrix} = (1)(-2) - (2)(3) = -2 - 6 = -8 \]

Putting it all together:
\[ \vec{A} \times \vec{B} = 8\hat{i} + 8\hat{j} - 8\hat{k} \]

### Step 4: Find the Magnitude of the Cross Product
Now we will find the magnitude of the vector \(\vec{A} \times \vec{B}\):
\[ |\vec{A} \times \vec{B}| = \sqrt{(8)^2 + (8)^2 + (-8)^2} = \sqrt{64 + 64 + 64} = \sqrt{192} = 8\sqrt{3} \]

### Step 5: Conclusion
The area of the parallelogram is given by the magnitude of the cross product:
\[ \text{Area} = 8\sqrt{3} \text{ square units} \]