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(j)+3hat(k)`

Answer

To determine the value of \( \lambda \) for which the vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) are coplanar, we need to use the condition that the scalar triple product of these vectors is equal to zero. This can be represented using the determinant of a matrix formed by the components of the vectors.

### Step-by-Step Solution:

1. **Identify the Vectors:**
- \( \vec{a} = \hat{i} + 3\hat{j} + \hat{k} \)
- \( \vec{b} = 2\hat{i} - \hat{j} - \hat{k} \)
- \( \vec{c} = \lambda \hat{j} + 3\hat{k} \)

2. **Write the Vectors in Component Form:**
- \( \vec{a} = (1, 3, 1) \)
- \( \vec{b} = (2, -1, -1) \)
- \( \vec{c} = (0, \lambda, 3) \)

3. **Set Up the Determinant:**
The condition for coplanarity is given by the determinant:
\[ \begin{vmatrix} 1 & 3 & 1 \\ 2 & -1 & -1 \\ 0 & \lambda & 3 \end{vmatrix} = 0 \]

4. **Calculate the Determinant:**
Expanding the determinant using the first row:
\[ = 1 \cdot \begin{vmatrix} -1 & -1 \\ \lambda & 3 \end{vmatrix} - 3 \cdot \begin{vmatrix} 2 & -1 \\ 0 & 3 \end{vmatrix} + 1 \cdot \begin{vmatrix} 2 & -1 \\ 0 & \lambda \end{vmatrix} \]

Now calculate each of the 2x2 determinants:
- First determinant:
\[ \begin{vmatrix} -1 & -1 \\ \lambda & 3 \end{vmatrix} = (-1)(3) - (-1)(\lambda) = -3 + \lambda = \lambda - 3 \]
- Second determinant:
\[ \begin{vmatrix} 2 & -1 \\ 0 & 3 \end{vmatrix} = (2)(3) - (-1)(0) = 6 \]
- Third determinant:
\[ \begin{vmatrix} 2 & -1 \\ 0 & \lambda \end{vmatrix} = (2)(\lambda) - (-1)(0) = 2\lambda \]

5. **Combine the Results:**
Now substituting back into the determinant:
\[ 1(\lambda - 3) - 3(6) + 1(2\lambda) = 0 \]
Simplifying:
\[ \lambda - 3 - 18 + 2\lambda = 0 \]
\[ 3\lambda - 21 = 0 \]

6. **Solve for \( \lambda \):**
\[ 3\lambda = 21 \implies \lambda = 7 \]

### Final Answer:
The value of \( \lambda \) for which the vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) are coplanar is \( \lambda = 7 \).