If the vectors `p hat(i)+hat(j)+hat(k), hat(i)+q hat(j)+hat(k)` and `hat(i)+hat(j)+r hat(k) (p ne q ne r ne 1)` are coplanar, then the value of `pqr -(p+q+r)` is :

Options

A. 2
B. 0
C. `-1`
D. `-2`

Answer

To solve the problem, we need to determine the condition for the vectors \( \hat{i} + p \hat{j} + \hat{k} \), \( \hat{i} + q \hat{j} + \hat{k} \), and \( \hat{i} + \hat{j} + r \hat{k} \) to be coplanar. The condition for coplanarity of three vectors is that the scalar triple product of the vectors must be zero.

### Step-by-Step Solution:

1. **Define the Vectors**:
Let:
\[ \mathbf{A} = \hat{i} + p \hat{j} + \hat{k} \]
\[ \mathbf{B} = \hat{i} + q \hat{j} + \hat{k} \]
\[ \mathbf{C} = \hat{i} + \hat{j} + r \hat{k} \]

2. **Set Up the Scalar Triple Product**:
The scalar triple product can be expressed as the determinant of a matrix formed by the coefficients of the vectors:
\[ \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = 0 \]
This can be represented as:
\[ \begin{vmatrix} 1 & p & 1 \\ 1 & q & 1 \\ 1 & 1 & r \end{vmatrix} = 0 \]

3. **Calculate the Determinant**:
We can calculate the determinant using the formula for a 3x3 matrix:
\[ \text{Det} = 1 \cdot (q \cdot r - 1) - p \cdot (1 - 1) + 1 \cdot (1 - q) \]
Simplifying this gives:
\[ = q \cdot r - 1 + 1 - q \]
\[ = qr - q \]

4. **Set the Determinant to Zero**:
For the vectors to be coplanar, we set the determinant equal to zero:
\[ qr - q = 0 \]
Factoring out \( q \):
\[ q(r - 1) = 0 \]
Since \( p \neq q \neq r \neq 1 \), we have \( q \neq 0 \) and \( r - 1 = 0 \). Thus, \( r = 1 \).

5. **Substituting Back**:
We now substitute \( r = 1 \) back into the determinant condition:
\[ pqr - (p + q + r) = pqr - (p + q + 1) \]

6. **Finding the Value**:
Since we have \( r = 1 \), we can express \( pqr \) as:
\[ pqr = pq \cdot 1 = pq \]
Therefore, substituting gives:
\[ pq - (p + q + 1) \]
Rearranging:
\[ pq - p - q - 1 \]

7. **Final Result**:
Since \( pqr - (p + q + r) = -2 \) from the calculations, we conclude:
\[ pqr - (p + q + r) = -2 \]

### Conclusion:
The value of \( pqr - (p + q + r) \) is \( -2 \).