Find x such that the four points `A\ (3,\ 2,\ 1)`, `B(4, x ,\ 5)`, `C\ (4,\ 2,\ 2)`and `D\ (6,\ 5,\ 1)`are coplanar

Explanation

To find the value of \( x \) such that the points \( A(3, 2, 1) \), \( B(4, x, 5) \), \( C(4, 2, 2) \), and \( D(6, 5, 1) \) are coplanar, we can use the concept of vectors and the scalar triple product. The points are coplanar if the scalar triple product of the vectors \( \overrightarrow{AB} \), \( \overrightarrow{AC} \), and \( \overrightarrow{AD} \) is zero.

### Step-by-step solution:

1. **Define the vectors**:
- \( \overrightarrow{A} = (3, 2, 1) \)
- \( \overrightarrow{B} = (4, x, 5) \)
- \( \overrightarrow{C} = (4, 2, 2) \)
...