The adjacent sides of a parallelogram are `vec(P) = 2 I - 3j + k and vec(Q) = -2i + 4j - k`. The are aof the parallelogram is `sqrtn` units. Find the value of n

Answer

To find the value of \( n \) in the area of the parallelogram formed by the vectors \( \vec{P} = 2\hat{i} - 3\hat{j} + \hat{k} \) and \( \vec{Q} = -2\hat{i} + 4\hat{j} - \hat{k} \), we will follow these steps:

### Step 1: Write down the vectors
The given vectors are:
\[ \vec{P} = 2\hat{i} - 3\hat{j} + \hat{k} \]
\[ \vec{Q} = -2\hat{i} + 4\hat{j} - \hat{k} \]

### Step 2: Compute the cross product \( \vec{P} \times \vec{Q} \)
The area of the parallelogram is given by the magnitude of the cross product of the two vectors. We will use the determinant method to find \( \vec{P} \times \vec{Q} \).

Set up the determinant:
\[ \vec{P} \times \vec{Q} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -3 & 1 \\ -2 & 4 & -1 \end{vmatrix} \]

### Step 3: Calculate the determinant
Expanding the determinant:
\[ \vec{P} \times \vec{Q} = \hat{i} \begin{vmatrix} -3 & 1 \\ 4 & -1 \end{vmatrix} - \hat{j} \begin{vmatrix} 2 & 1 \\ -2 & -1 \end{vmatrix} + \hat{k} \begin{vmatrix} 2 & -3 \\ -2 & 4 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
1. \( \begin{vmatrix} -3 & 1 \\ 4 & -1 \end{vmatrix} = (-3)(-1) - (1)(4) = 3 - 4 = -1 \)

2. \( \begin{vmatrix} 2 & 1 \\ -2 & -1 \end{vmatrix} = (2)(-1) - (1)(-2) = -2 + 2 = 0 \)

3. \( \begin{vmatrix} 2 & -3 \\ -2 & 4 \end{vmatrix} = (2)(4) - (-3)(-2) = 8 - 6 = 2 \)

Putting it all together:
\[ \vec{P} \times \vec{Q} = -1\hat{i} - 0\hat{j} + 2\hat{k} = -\hat{i} + 2\hat{k} \]

### Step 4: Find the magnitude of the cross product
The magnitude of \( \vec{P} \times \vec{Q} \) is given by:
\[ |\vec{P} \times \vec{Q}| = \sqrt{(-1)^2 + 0^2 + 2^2} = \sqrt{1 + 0 + 4} = \sqrt{5} \]

### Step 5: Relate the area to \( n \)
The area of the parallelogram is equal to the magnitude of the cross product:
\[ \text{Area} = |\vec{P} \times \vec{Q}| = \sqrt{5} \]
We are given that the area is \( \sqrt{n} \). Therefore, we can equate:
\[ \sqrt{n} = \sqrt{5} \]

### Step 6: Solve for \( n \)
Squaring both sides gives:
\[ n = 5 \]

### Final Answer
The value of \( n \) is \( 5 \).

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