PQRS is a parallelogram, where `vec(PQ) = 3hat(i) + 2hat(j) - m hat(k) and vec(PS) = hat(i) + 3hat(j) + hat(k)` and the area of the parallelogram is `sqrt90`. What is the value of m?

Options

A. 1
B. `-1`
C. 2
D. `-2`

Answer

To solve the problem, we need to find the value of \( m \) in the parallelogram \( PQRS \) given the vectors \( \vec{PQ} \) and \( \vec{PS} \) and the area of the parallelogram.

### Step-by-Step Solution:

1. **Identify the Given Vectors:**
\[ \vec{PQ} = 3\hat{i} + 2\hat{j} - m\hat{k} \]
\[ \vec{PS} = \hat{i} + 3\hat{j} + \hat{k} \]

2. **Calculate the Cross Product \( \vec{PQ} \times \vec{PS} \):**
The area of the parallelogram can be calculated using the magnitude of the cross product of the two vectors:
\[ \text{Area} = |\vec{PQ} \times \vec{PS}| \]

To compute the cross product, we can use the determinant:
\[ \vec{PQ} \times \vec{PS} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 2 & -m \\ 1 & 3 & 1 \end{vmatrix} \]

3. **Expand the Determinant:**
\[ \vec{PQ} \times \vec{PS} = \hat{i} \begin{vmatrix} 2 & -m \\ 3 & 1 \end{vmatrix} - \hat{j} \begin{vmatrix} 3 & -m \\ 1 & 1 \end{vmatrix} + \hat{k} \begin{vmatrix} 3 & 2 \\ 1 & 3 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
- For \( \hat{i} \):
\[ 2 \cdot 1 - (-m) \cdot 3 = 2 + 3m \]
- For \( \hat{j} \):
\[ 3 \cdot 1 - (-m) \cdot 1 = 3 + m \]
- For \( \hat{k} \):
\[ 3 \cdot 3 - 2 \cdot 1 = 9 - 2 = 7 \]

Thus, we have:
\[ \vec{PQ} \times \vec{PS} = (2 + 3m)\hat{i} - (3 + m)\hat{j} + 7\hat{k} \]

4. **Magnitude of the Cross Product:**
The magnitude of the vector is given by:
\[ |\vec{PQ} \times \vec{PS}| = \sqrt{(2 + 3m)^2 + (3 + m)^2 + 7^2} \]

Setting this equal to the area:
\[ \sqrt{(2 + 3m)^2 + (3 + m)^2 + 49} = \sqrt{90} \]

5. **Square Both Sides:**
\[ (2 + 3m)^2 + (3 + m)^2 + 49 = 90 \]

6. **Simplify the Equation:**
Expanding the squares:
\[ (2 + 3m)^2 = 4 + 12m + 9m^2 \]
\[ (3 + m)^2 = 9 + 6m + m^2 \]
Thus, we have:
\[ 4 + 12m + 9m^2 + 9 + 6m + m^2 + 49 = 90 \]
Combining like terms:
\[ 10m^2 + 18m + 62 = 90 \]
Rearranging gives:
\[ 10m^2 + 18m - 28 = 0 \]

7. **Divide by 2:**
\[ 5m^2 + 9m - 14 = 0 \]

8. **Factor the Quadratic:**
\[ (5m - 2)(m + 7) = 0 \]
This gives us:
\[ 5m - 2 = 0 \quad \text{or} \quad m + 7 = 0 \]
Thus:
\[ m = \frac{2}{5} \quad \text{or} \quad m = -7 \]

9. **Select the Valid Solution:**
Since \( m \) must be a valid physical quantity, we take:
\[ m = \frac{2}{5} \]

### Final Answer:
The value of \( m \) is \( \frac{2}{5} \).