Three vertices of a parallelogram ABCD are `A(3,-1,2),\ B(1,2,-4)a n d\ C(-1,1,2)dot` Find the coordinates of the fourth vertex.

Explanation

To find the coordinates of the fourth vertex \( D(x, y, z) \) of the parallelogram \( ABCD \) given the vertices \( A(3, -1, 2) \), \( B(1, 2, -4) \), and \( C(-1, 1, 2) \), we can use the property that the diagonals of a parallelogram bisect each other.

### Step-by-Step Solution:

1. **Identify the Midpoint of Diagonal AC:**
The midpoint \( M_{AC} \) of diagonal \( AC \) can be calculated using the midpoint formula:
\[
M_{AC} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)
...