For what value of `'lambda'` are the following vectors coplanar ?
`vec(a)=hat(i)+3hat(j)+hat(k), vec(b)=2hat(i)-hat(j)-hat(k)` and `vec(c )=lambda hat(i)+7hat(j)+3hat(k)`.

Answer

To determine the value of \( \lambda \) for which the vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) are coplanar, we can use the condition that the scalar triple product of the vectors must be zero. The scalar triple product can be calculated using the determinant of a matrix formed by the components of the vectors.

### Step-by-step Solution:

1. **Identify the vectors**:
\[ \vec{a} = \hat{i} + 3\hat{j} + \hat{k} \quad \text{(coefficients: 1, 3, 1)} \]
\[ \vec{b} = 2\hat{i} - \hat{j} - \hat{k} \quad \text{(coefficients: 2, -1, -1)} \]
\[ \vec{c} = \lambda \hat{i} + 7\hat{j} + 3\hat{k} \quad \text{(coefficients: } \lambda, 7, 3\text{)} \]

2. **Set up the determinant**:
The scalar triple product can be expressed as:
\[ \text{det} \begin{pmatrix} 1 & 3 & 1 \\ 2 & -1 & -1 \\ \lambda & 7 & 3 \end{pmatrix} \]
We need to set this determinant equal to zero for coplanarity.

3. **Calculate the determinant**:
Expanding the determinant:
\[ = 1 \cdot \begin{vmatrix} -1 & -1 \\ 7 & 3 \end{vmatrix} - 3 \cdot \begin{vmatrix} 2 & -1 \\ \lambda & 3 \end{vmatrix} + 1 \cdot \begin{vmatrix} 2 & -1 \\ \lambda & 7 \end{vmatrix} \]

Now calculating each of these 2x2 determinants:
- For the first determinant:
\[ \begin{vmatrix} -1 & -1 \\ 7 & 3 \end{vmatrix} = (-1)(3) - (-1)(7) = -3 + 7 = 4 \]
- For the second determinant:
\[ \begin{vmatrix} 2 & -1 \\ \lambda & 3 \end{vmatrix} = (2)(3) - (-1)(\lambda) = 6 + \lambda = 6 + \lambda \]
- For the third determinant:
\[ \begin{vmatrix} 2 & -1 \\ \lambda & 7 \end{vmatrix} = (2)(7) - (-1)(\lambda) = 14 + \lambda \]

4. **Substituting back into the determinant**:
Now substituting back into the determinant:
\[ = 1 \cdot 4 - 3(6 + \lambda) + 1(14 + \lambda) \]
Simplifying this:
\[ = 4 - 18 - 3\lambda + 14 + \lambda \]
\[ = 4 - 18 + 14 - 3\lambda + \lambda \]
\[ = 0 - 2\lambda \]

5. **Setting the determinant to zero**:
For the vectors to be coplanar, we set the expression equal to zero:
\[ -2\lambda = 0 \]
Solving for \( \lambda \):
\[ \lambda = 0 \]

### Conclusion:
The value of \( \lambda \) for which the vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) are coplanar is:
\[ \lambda = 0 \]