. Suppose you have a parallelogram with vertices P, Q, R, and S so that the displacement
vector from P to Q equals the displacement vector from S to R. If P = (2, 0, 1), Q = (3, 1, 0)
and R = (4, 3, 5) find the coordinates of the point, S. Draw a schematic diagram showing
how you arrived at your answer.


### Problem Analysis
Given the vertices \( P \), \( Q \), and \( R \) of a parallelogram, you need to find the coordinates of the fourth vertex \( S \). In a parallelogram, opposite sides are equal in magnitude and direction. Therefore, the displacement vector from \( P \) to \( Q \) should be equal to the displacement vector from \( S \) to \( R \).

### Step-by-Step Solution

1. **Find the displacement vector \( \overrightarrow{PQ} \):**
   \[
   \overrightarrow{PQ} = Q - P = (3, 1, 0) - (2, 0, 1) = (1, 1, -1)
   \]

2. **Set the displacement vector \( \overrightarrow{SR} \) equal to \( \overrightarrow{PQ} \):**
   \[
   \overrightarrow{SR} = R - S
   \]
   Since \( \overrightarrow{SR} = \overrightarrow{PQ} \), we have:
   \[
   R - S = \overrightarrow{PQ}
   \]

3. **Find \( S \):**
   \[
   S = R - \overrightarrow{PQ} = (4, 3, 5) - (1, 1, -1) = (3, 2, 6)
   \]
   So, the coordinates of point \( S \) are \( (3, 2, 6) \).

### Schematic Diagram

I’ll describe how to sketch the diagram:
- Start by plotting the points \( P = (2, 0, 1) \), \( Q = (3, 1, 0) \), and \( R = (4, 3, 5) \) in a 3D coordinate system.
- Draw the vector \( \overrightarrow{PQ} \) from \( P \) to \( Q \).
- Next, draw the vector \( \overrightarrow{SR} \) starting from \( S \) (which you now know is \( (3, 2, 6) \)) to \( R \).
- The vectors \( \overrightarrow{PQ} \) and \( \overrightarrow{SR} \) should be parallel and equal in magnitude, forming the parallelogram.

### Final Answer
The coordinates of the point \( S \) are \( \boxed{(3, 2, 6)} \).

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Do you want more details or have any questions?

**Related Questions:**
1. How can you prove the property of parallelograms using vectors?
2. What is the formula for the area of a parallelogram in vector form?
3. How would the solution change if the points were given in a different dimension?
4. Can you generalize the concept to find the missing vertex of any quadrilateral?
5. How does the concept of vector addition apply to this problem?

**Tip:** Always remember that opposite sides of a parallelogram are not only equal in length but also have the same direction. This makes vector addition a powerful tool in geometry.

Learn how to find the coordinates of point S in a parallelogram with given vertices P, Q, and R using vector displacement. Detailed step-by-step solution included. Suitable for Grades 10-12.

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