Area of parallelogram whose adjacent sides of `a = i +2j+3k, b = 3i-2j+k` is

Options

A. `5sqrt(2)`
B. `8sqrt(3)`
C. 6
D. none

Answer

To find the area of the parallelogram formed by the vectors \( \mathbf{a} \) and \( \mathbf{b} \), we can use the formula for the area, which is given by the magnitude of the cross product of the two vectors.

Given:
\[ \mathbf{a} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \]
\[ \mathbf{b} = 3\mathbf{i} - 2\mathbf{j} + \mathbf{k} \]

### Step 1: Calculate the cross product \( \mathbf{a} \times \mathbf{b} \)

The cross product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) can be calculated using the determinant of a matrix formed by the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) and the components of the vectors \( \mathbf{a} \) and \( \mathbf{b} \).

\[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 2 & 3 \\ 3 & -2 & 1 \end{vmatrix} \]

### Step 2: Calculate the determinant

Now we will calculate the determinant:

\[ \mathbf{a} \times \mathbf{b} = \mathbf{i} \begin{vmatrix} 2 & 3 \\ -2 & 1 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 1 & 3 \\ 3 & 1 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 1 & 2 \\ 3 & -2 \end{vmatrix} \]

Calculating each of the 2x2 determinants:

1. \( \begin{vmatrix} 2 & 3 \\ -2 & 1 \end{vmatrix} = (2)(1) - (3)(-2) = 2 + 6 = 8 \)
2. \( \begin{vmatrix} 1 & 3 \\ 3 & 1 \end{vmatrix} = (1)(1) - (3)(3) = 1 - 9 = -8 \)
3. \( \begin{vmatrix} 1 & 2 \\ 3 & -2 \end{vmatrix} = (1)(-2) - (2)(3) = -2 - 6 = -8 \)

Putting it all together:

\[ \mathbf{a} \times \mathbf{b} = \mathbf{i}(8) - \mathbf{j}(-8) + \mathbf{k}(-8) \]
\[ = 8\mathbf{i} + 8\mathbf{j} - 8\mathbf{k} \]

### Step 3: Calculate the magnitude of the cross product

The magnitude of the vector \( \mathbf{a} \times \mathbf{b} \) is given by:

\[ |\mathbf{a} \times \mathbf{b}| = \sqrt{(8)^2 + (8)^2 + (-8)^2} \]
\[ = \sqrt{64 + 64 + 64} = \sqrt{192} = 8\sqrt{3} \]

### Step 4: Conclusion

The area of the parallelogram is equal to the magnitude of the cross product:

\[ \text{Area} = |\mathbf{a} \times \mathbf{b}| = 8\sqrt{3} \]