If the three successive vertices of a parallelogram have the position vectors as, A(–3, –2, 0), B(3, –3, 1) and C(5, 0, 2). Then find
Position vector of the fourth vertex D

Answer

To find the position vector of the fourth vertex \( D \) of the parallelogram given the position vectors of the vertices \( A \), \( B \), and \( C \), we can use the property that the diagonals of a parallelogram bisect each other.

### Step-by-Step Solution:

1. **Identify the Position Vectors**:
- Let \( \vec{A} = (-3, -2, 0) \)
- Let \( \vec{B} = (3, -3, 1) \)
- Let \( \vec{C} = (5, 0, 2) \)

2. **Find the Midpoint of Diagonal AC**:
The midpoint \( O \) of diagonal \( AC \) can be found using the midpoint formula:
\[ \vec{O} = \frac{\vec{A} + \vec{C}}{2} \]
Substituting the values:
\[ \vec{O} = \frac{(-3, -2, 0) + (5, 0, 2)}{2} = \frac{(-3 + 5, -2 + 0, 0 + 2)}{2} = \frac{(2, -2, 2)}{2} = (1, -1, 1) \]

3. **Use the Midpoint to Find Position Vector of D**:
Since \( O \) is also the midpoint of diagonal \( BD \), we can express this as:
\[ \vec{O} = \frac{\vec{B} + \vec{D}}{2} \]
Rearranging gives:
\[ \vec{D} = 2\vec{O} - \vec{B} \]
Substituting the values of \( \vec{O} \) and \( \vec{B} \):
\[ \vec{D} = 2(1, -1, 1) - (3, -3, 1) = (2, -2, 2) - (3, -3, 1) = (2 - 3, -2 + 3, 2 - 1) = (-1, 1, 1) \]

4. **Final Result**:
The position vector of the fourth vertex \( D \) is:
\[ \vec{D} = (-1, 1, 1) \]