The area of the parallenlogram determined by two adjacentt sides as `A=2hat(i)+hat(j)-3hat(k)andB=12hat(j)-2hat(k)` is approximately

Options

A. 43
B. 56
C. 38
D. 74

Explanation

To find the area of the parallelogram determined by the vectors \( \mathbf{A} \) and \( \mathbf{B} \), we will use the formula for the area, which is given by the magnitude of the cross product of the two vectors.

### Step 1: Define the vectors
The vectors are given as:
\[ \mathbf{A} = 2\hat{i} + \hat{j} - 3\hat{k} \]
\[ \mathbf{B} = 12\hat{j} - 2\hat{k} \]

### Step 2: Calculate the cross product \( \mathbf{A} \times \mathbf{B} \)
To find the cross product, we can use the determinant of a matrix formed by the unit vectors \( \hat{i}, \hat{j}, \hat{k} \) and the components of the vectors \( \mathbf{A} \) and \( \mathbf{B} \):

\[ \mathbf{A} \times \mathbf{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & -3 \\ 0 & 12 & -2 \end{vmatrix} \]

### Step 3: Calculate the determinant
Expanding this determinant, we have:

\[ \mathbf{A} \times \mathbf{B} = \hat{i} \begin{vmatrix} 1 & -3 \\ 12 & -2 \end{vmatrix} - \hat{j} \begin{vmatrix} 2 & -3 \\ 0 & -2 \end{vmatrix} + \hat{k} \begin{vmatrix} 2 & 1 \\ 0 & 12 \end{vmatrix} \]

Calculating each of these 2x2 determinants:

1. For \( \hat{i} \):
\[ \begin{vmatrix} 1 & -3 \\ 12 & -2 \end{vmatrix} = (1)(-2) - (-3)(12) = -2 + 36 = 34 \]

2. For \( \hat{j} \):
\[ \begin{vmatrix} 2 & -3 \\ 0 & -2 \end{vmatrix} = (2)(-2) - (0)(-3) = -4 \]

3. For \( \hat{k} \):
\[ \begin{vmatrix} 2 & 1 \\ 0 & 12 \end{vmatrix} = (2)(12) - (0)(1) = 24 \]

Putting it all together:
\[ \mathbf{A} \times \mathbf{B} = 34\hat{i} + 4\hat{j} + 24\hat{k} \]

### Step 4: Find the magnitude of the cross product
Now we find the magnitude of \( \mathbf{A} \times \mathbf{B} \):
\[ |\mathbf{A} \times \mathbf{B}| = \sqrt{(34)^2 + (4)^2 + (24)^2} \]
Calculating each term:
\[ (34)^2 = 1156, \quad (4)^2 = 16, \quad (24)^2 = 576 \]
Adding these:
\[ 1156 + 16 + 576 = 1748 \]
Thus,
\[ |\mathbf{A} \times \mathbf{B}| = \sqrt{1748} \]

### Step 5: Calculate the square root
Calculating the square root:
\[ \sqrt{1748} \approx 41.8 \]

### Conclusion
The area of the parallelogram determined by the vectors \( \mathbf{A} \) and \( \mathbf{B} \) is approximately \( 41.8 \).

---