Let P, Q, R and S be the points on the plane with position vectors `-2hat(i)-hat(j), 4hat(i), 3hat(i)+3hat(j) and -3hat(i)+2hat(j)`, respectively. The quadrilateral PQRS must be

Options

A. parallelogram, which is neither a rhombus nor a rectangle
B. square
C. rectangle, but not a square
D. rhombus, but not a square

Answer

To determine the type of quadrilateral formed by the points P, Q, R, and S with given position vectors, we will follow these steps:

### Step 1: Identify the position vectors
The position vectors of the points are:
- \( \vec{P} = -2\hat{i} - \hat{j} \)
- \( \vec{Q} = 4\hat{i} \)
- \( \vec{R} = 3\hat{i} + 3\hat{j} \)
- \( \vec{S} = -3\hat{i} + 2\hat{j} \)

### Step 2: Calculate the vectors for the sides of the quadrilateral
1. **Vector \( \vec{PQ} \)**:
\[ \vec{PQ} = \vec{Q} - \vec{P} = (4\hat{i}) - (-2\hat{i} - \hat{j}) = 4\hat{i} + 2\hat{i} + \hat{j} = 6\hat{i} + \hat{j} \]

2. **Vector \( \vec{QR} \)**:
\[ \vec{QR} = \vec{R} - \vec{Q} = (3\hat{i} + 3\hat{j}) - (4\hat{i}) = -\hat{i} + 3\hat{j} \]

3. **Vector \( \vec{RS} \)**:
\[ \vec{RS} = \vec{S} - \vec{R} = (-3\hat{i} + 2\hat{j}) - (3\hat{i} + 3\hat{j}) = -6\hat{i} - \hat{j} \]

4. **Vector \( \vec{SP} \)**:
\[ \vec{SP} = \vec{P} - \vec{S} = (-2\hat{i} - \hat{j}) - (-3\hat{i} + 2\hat{j}) = \hat{i} - 3\hat{j} \]

### Step 3: Calculate the magnitudes of the vectors
1. **Magnitude of \( \vec{PQ} \)**:
\[ |\vec{PQ}| = \sqrt{(6)^2 + (1)^2} = \sqrt{36 + 1} = \sqrt{37} \]

2. **Magnitude of \( \vec{QR} \)**:
\[ |\vec{QR}| = \sqrt{(-1)^2 + (3)^2} = \sqrt{1 + 9} = \sqrt{10} \]

3. **Magnitude of \( \vec{RS} \)**:
\[ |\vec{RS}| = \sqrt{(-6)^2 + (-1)^2} = \sqrt{36 + 1} = \sqrt{37} \]

4. **Magnitude of \( \vec{SP} \)**:
\[ |\vec{SP}| = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]

### Step 4: Check the conditions for a parallelogram
We have:
- \( |\vec{PQ}| = |\vec{RS}| \) (both equal to \( \sqrt{37} \))
- \( |\vec{QR}| = |\vec{SP}| \) (both equal to \( \sqrt{10} \))

Since opposite sides are equal, this indicates that the quadrilateral is at least a parallelogram.

### Step 5: Check if it is a rectangle
To check if it is a rectangle, we need to see if adjacent sides are perpendicular. We can do this by calculating the dot product of \( \vec{PQ} \) and \( \vec{QR} \):

\[ \vec{PQ} \cdot \vec{QR} = (6\hat{i} + \hat{j}) \cdot (-\hat{i} + 3\hat{j}) = 6(-1) + 1(3) = -6 + 3 = -3 \]

Since the dot product is non-zero, \( \vec{PQ} \) is not perpendicular to \( \vec{QR} \).

### Conclusion
Since opposite sides are equal and adjacent sides are not perpendicular, the quadrilateral PQRS is a parallelogram but not a rectangle or a rhombus.

### Final Answer
The quadrilateral PQRS is a **parallelogram**.

---