The number of real values of a for which the vectors `hati+2hatj+hatk, ahati+hatj+2hatk` and `hati+2hatj+ahatk` are coplanar is

Options

A. 1
B. 2
C. 3
D. 0

Explanation

To determine the number of real values of \( a \) for which the vectors \( \hat{i} + 2\hat{j} + \hat{k} \), \( a\hat{i} + \hat{j} + 2\hat{k} \), and \( \hat{i} + 2\hat{j} + a\hat{k} \) are coplanar, we can use the condition that the scalar triple product of these vectors must be zero.

### Step-by-step Solution:

1. **Write the Vectors**:
Let:
\[
\mathbf{A} = \hat{i} + 2\hat{j} + \hat{k}
...