The value of ‘`lambda` ’ so that the vectors `hati - 3̂ hatj + hatk , 2̂ hati + lambda hatj + hatk ` and `3 hati + hatj - 2 hatk` are coplanar, will be

Options

A. 0
B. 2
C. `-(1)/(2)`
D. `-4`

Answer

To find the value of \( \lambda \) such that the vectors \( \hat{i} - 3\hat{j} + \hat{k} \), \( 2\hat{i} + \lambda \hat{j} + \hat{k} \), and \( 3\hat{i} + \hat{j} - 2\hat{k} \) are coplanar, we can use the property that three vectors are coplanar if the scalar triple product of the vectors is zero. This can be determined using the determinant of a matrix formed by the vectors.

### Step-by-Step Solution:

1. **Identify the Vectors**:
The vectors are:
- \( \mathbf{A} = \hat{i} - 3\hat{j} + \hat{k} \)
- \( \mathbf{B} = 2\hat{i} + \lambda \hat{j} + \hat{k} \)
- \( \mathbf{C} = 3\hat{i} + \hat{j} - 2\hat{k} \)

2. **Write the Coefficients**:
The coefficients of \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) for each vector are:
- For \( \mathbf{A} \): \( (1, -3, 1) \)
- For \( \mathbf{B} \): \( (2, \lambda, 1) \)
- For \( \mathbf{C} \): \( (3, 1, -2) \)

3. **Set Up the Determinant**:
We need to set up the determinant of the matrix formed by these vectors:
\[ \begin{vmatrix} 1 & -3 & 1 \\ 2 & \lambda & 1 \\ 3 & 1 & -2 \end{vmatrix} = 0 \]

4. **Calculate the Determinant**:
The determinant can be calculated as follows:
\[ = 1 \cdot \begin{vmatrix} \lambda & 1 \\ 1 & -2 \end{vmatrix} - (-3) \cdot \begin{vmatrix} 2 & 1 \\ 3 & -2 \end{vmatrix} + 1 \cdot \begin{vmatrix} 2 & \lambda \\ 3 & 1 \end{vmatrix} \]

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

- \( \begin{vmatrix} 2 & 1 \\ 3 & -2 \end{vmatrix} = 2 \cdot (-2) - 1 \cdot 3 = -4 - 3 = -7 \)

- \( \begin{vmatrix} 2 & \lambda \\ 3 & 1 \end{vmatrix} = 2 \cdot 1 - \lambda \cdot 3 = 2 - 3\lambda \)

Plugging these back into the determinant:
\[ 1(-2\lambda - 1) + 3(-7) + 1(2 - 3\lambda) = 0 \]
This simplifies to:
\[ -2\lambda - 1 - 21 + 2 - 3\lambda = 0 \]
Combining like terms:
\[ -5\lambda - 20 = 0 \]

5. **Solve for \( \lambda \)**:
\[ -5\lambda = 20 \implies \lambda = -4 \]

### Final Answer:
The value of \( \lambda \) is \( -4 \).