The co-ordinates of the vertices of a parallelogram ABCD are A(-1,2,3), B(2, -4,1) and C(1,2,-1). Find the co-ordinates of its 4th vertex.

Explanation

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

### Step-by-step Solution:

1. **Identify the coordinates of the given points**:
- \( A(-1, 2, 3) \)
- \( B(2, -4, 1) \)
- \( C(1, 2, -1) \)

2. **Let the coordinates of point \( D \) be \( D(x, y, z) \)**.

3. **Use the property of the diagonals**:
The midpoints of the diagonals \( AC \) and \( BD \) should be equal.
- The midpoint of \( AC \) is given by:
\[ M_{AC} = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2}, \frac{z_A + z_C}{2} \right) = \left( \frac{-1 + 1}{2}, \frac{2 + 2}{2}, \frac{3 - 1}{2} \right) = \left( 0, 2, 1 \right) \]

- The midpoint of \( BD \) is given by:
\[ M_{BD} = \left( \frac{x_B + x_D}{2}, \frac{y_B + y_D}{2}, \frac{z_B + z_D}{2} \right) = \left( \frac{2 + x}{2}, \frac{-4 + y}{2}, \frac{1 + z}{2} \right) \]

4. **Set the midpoints equal**:
Since \( M_{AC} = M_{BD} \), we can set up the following equations:
- For the x-coordinates:
\[ \frac{2 + x}{2} = 0 \implies 2 + x = 0 \implies x = -2 \]
- For the y-coordinates:
\[ \frac{-4 + y}{2} = 2 \implies -4 + y = 4 \implies y = 8 \]
- For the z-coordinates:
\[ \frac{1 + z}{2} = 1 \implies 1 + z = 2 \implies z = 1 \]

5. **Combine the results**:
The coordinates of point \( D \) are \( D(-2, 8, 1) \).

### Final Answer:
The coordinates of the fourth vertex \( D \) are \( D(-2, 8, 1) \).