Find `lambda` if the vectors ` veca= hat i+3\ hat j+ hat k , vecb=2 hat i- hat j- hat k\ and\ vecc=lambda hat i+7\ hatj+3 hat k`are coplanar.

Explanation

To find the value of \( \lambda \) such that the vectors \( \vec{a} = \hat{i} + 3\hat{j} + \hat{k} \), \( \vec{b} = 2\hat{i} - \hat{j} - \hat{k} \), and \( \vec{c} = \lambda \hat{i} + 7\hat{j} + 3\hat{k} \) are coplanar, we can use the scalar triple product. The vectors are coplanar if the scalar triple product is equal to zero.

### Step-by-Step Solution:

1. **Set up the vectors**:
\[ \vec{a} = \hat{i} + 3\hat{j} + \hat{k} \]
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