Find `lambda` so that the vectors ` vec a=2 hat i- hat j+ hat k ,\ vec b= hat i+2 hat j-3 hat k\ a n d\ vec c=3 hat i+lambda hat j+5 hat k` re coplanar.

Explanation

To find the value of \( \lambda \) such that the vectors \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \), \( \vec{b} = \hat{i} + 2\hat{j} - 3\hat{k} \), and \( \vec{c} = 3\hat{i} + \lambda\hat{j} + 5\hat{k} \) are coplanar, we can use the condition that the scalar triple product of the vectors must be zero.

### Step-by-Step Solution:

1. **Set up the Scalar Triple Product**:
The vectors are coplanar if the scalar triple product \( \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \). We can express this as a determinant:
\[
\begin{vmatrix}
...