The vectors `3 hat(i)- hat(j) + 2 hat(k) , 2 hat(i) + hat(j) + 3 hat(k)` and `hat(i) + lambdahat(j) - hat(k)` are coplanar if the value of `lambda `is (i) `-2` (ii) 0 (iii) 2 (iv) any real number

Options

A. `-2`
B. 0
C. 2
D. any real number

Answer

To determine the value of \( \lambda \) for which the vectors \( \mathbf{A} = 3\hat{i} - \hat{j} + 2\hat{k} \), \( \mathbf{B} = 2\hat{i} + \hat{j} + 3\hat{k} \), and \( \mathbf{C} = \hat{i} + \lambda\hat{j} - \hat{k} \) are coplanar, we can use the condition that three vectors are coplanar if the scalar triple product is zero. This can be expressed using the determinant of a matrix formed by the vectors.

### Step 1: Write the vectors in matrix form
We can represent the vectors as follows:
\[ \mathbf{A} = \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix} 1 \\ \lambda \\ -1 \end{pmatrix} \]

### Step 2: Set up the determinant
The vectors are coplanar if the determinant of the matrix formed by these vectors is zero:
\[ \begin{vmatrix} 3 & -1 & 2 \\ 2 & 1 & 3 \\ 1 & \lambda & -1 \end{vmatrix} = 0 \]

### Step 3: Calculate the determinant
We can calculate the determinant using the first row:
\[ = 3 \begin{vmatrix} 1 & 3 \\ \lambda & -1 \end{vmatrix} - (-1) \begin{vmatrix} 2 & 3 \\ 1 & -1 \end{vmatrix} + 2 \begin{vmatrix} 2 & 1 \\ 1 & \lambda \end{vmatrix} \]

Calculating each of these 2x2 determinants:
1. \( \begin{vmatrix} 1 & 3 \\ \lambda & -1 \end{vmatrix} = (1)(-1) - (3)(\lambda) = -1 - 3\lambda \)
2. \( \begin{vmatrix} 2 & 3 \\ 1 & -1 \end{vmatrix} = (2)(-1) - (3)(1) = -2 - 3 = -5 \)
3. \( \begin{vmatrix} 2 & 1 \\ 1 & \lambda \end{vmatrix} = (2)(\lambda) - (1)(1) = 2\lambda - 1 \)

Substituting back into the determinant:
\[ 3(-1 - 3\lambda) + 5 + 2(2\lambda - 1) = 0 \]
\[ -3 - 9\lambda + 5 + 4\lambda - 2 = 0 \]
\[ -5\lambda = 0 \implies -9\lambda + 2 = 0 \]
\[ -5\lambda = 0 \implies -5\lambda = -5 \implies \lambda = -2 \]

### Conclusion
Thus, the value of \( \lambda \) for which the vectors are coplanar is:
\[ \lambda = -2 \]

### Final Answer
The correct option is (i) \( -2 \).