Determine `lambda` such that a vector `vec(r)` is at right angles to each of the vectors
`vec(a) = lambda hat(i) + hat(j) + 3hat(k), vec(b) = 2hat(i) + hat(j) - lambda hat(k), vec(c) = -2hat(i) + lambda hat(j) + 3hat(k)`.

Answer

To determine the value of `lambda` such that the vector `vec(r)` is at right angles to each of the vectors `vec(a)`, `vec(b)`, and `vec(c)`, we can use the concept of coplanarity of vectors. If `vec(r)` is orthogonal to all three vectors, then the vectors are coplanar, which means the determinant of the matrix formed by these vectors should be equal to zero.

### Step-by-step Solution:

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

2. **Set up the Determinant**:
The determinant of the matrix formed by these vectors should be zero for the vectors to be coplanar:
\[ \begin{vmatrix} \lambda & 1 & 3 \\ 2 & 1 & -\lambda \\ -2 & \lambda & 3 \end{vmatrix} = 0 \]

3. **Calculate the Determinant**:
We can calculate the determinant using the formula for a 3x3 matrix:
\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]
Applying this to our matrix:
\[ D = \lambda(1 \cdot 3 - (-\lambda) \cdot \lambda) - 1(2 \cdot 3 - (-\lambda)(-2)) + 3(2 \cdot \lambda - 1 \cdot (-2)) \]
Simplifying each term:
- First term: \( \lambda(3 + \lambda^2) \)
- Second term: \( -1(6 - 2\lambda) = -6 + 2\lambda \)
- Third term: \( 3(2\lambda + 2) = 6\lambda + 6 \)

Now, combine these:
\[ D = \lambda(3 + \lambda^2) - 6 + 2\lambda + 6\lambda + 6 \]
\[ D = \lambda^3 + 3\lambda + 6\lambda - 6 + 6 \]
\[ D = \lambda^3 + 11\lambda \]

4. **Set the Determinant to Zero**:
For coplanarity:
\[ \lambda^3 + 11\lambda = 0 \]

5. **Factor the Equation**:
Factor out `lambda`:
\[ \lambda(\lambda^2 + 11) = 0 \]

6. **Solve for `lambda`**:
This gives us:
- \( \lambda = 0 \)
- \( \lambda^2 + 11 = 0 \) (which has no real solutions)

Thus, the only real solution is:
\[ \lambda = 0 \]

### Final Answer:
The value of `lambda` is \( \lambda = 0 \).