The vectors `lamdahati+hatj+2hatk,hati+lamdahatj-hatkand2hati-hatj+lamdahatk` are coplanar, if

Options

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

Explanation

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

### Step-by-Step Solution:

1. **Write the vectors in component form:**
\[ \vec{A} = \begin{pmatrix} \lambda \\ 1 \\ 2 \end{pmatrix}, \quad \vec{B} = \begin{pmatrix} 1 \\ \lambda \\ -1 \end{pmatrix}, \quad \vec{C} = \begin{pmatrix} 2 \\ -1 \\ \lambda \end{pmatrix} \]
...