Find `lambda`if the vectors ` hati\ -\ hatj\ +\ hatk,\ 3 hati+\ hatj\ +\ 2 hatk\ and\ hati+lambda hatj+ hat3k\ `are coplanar

Explanation

To find the value of \( \lambda \) such that the vectors \( \hat{i} - \hat{j} + \hat{k} \), \( 3\hat{i} + \hat{j} + 2\hat{k} \), and \( \hat{i} + \lambda \hat{j} + 3\hat{k} \) are coplanar, we need to use the condition that the scalar triple product of these vectors is zero.

### Step-by-Step Solution:

1. **Define the Vectors:**
Let:
\[
\mathbf{a} = \hat{i} - \hat{j} + \hat{k} = (1, -1, 1)
...