Priya says that the sum of two vectors by the parallelogram method is `vecR=5hati`. Subhangi says it is `vecR=hati+4hatj`. Both used the parallelogram method, but one used the wrong diagonal. Which one of the vector pairs below contains the original two vectors.?

Options

A. `vecA=+3hati-2hatj, vecB=-2hati+2hatj`
B. `vecA=-3hati-2hatj, vecB=+2hati+2hatj`
C. `vecA=+3hati+2hatj, vecB=-2hati-2hatj`
D. `vecA=+3hati+2hatj, vecB=-2hati+2hatj`

Answer

To solve the problem, we need to determine which pair of vectors corresponds to the two vectors that Priya and Subhangi used to arrive at their respective results for the resultant vector using the parallelogram method.

### Step-by-Step Solution

1. **Understanding the Resultant Vectors**:
- Priya's resultant vector is \( \vec{R} = 5 \hat{i} \).
- Subhangi's resultant vector is \( \vec{R} = \hat{i} + 4 \hat{j} \).

2. **Using the Parallelogram Law**:
- The parallelogram law states that if two vectors \( \vec{A} \) and \( \vec{B} \) are represented as two adjacent sides of a parallelogram, then the diagonal represents the resultant vector \( \vec{R} \).
- Mathematically, \( \vec{R} = \vec{A} + \vec{B} \).

3. **Analyzing the Given Resultants**:
- We need to find two vectors \( \vec{A} \) and \( \vec{B} \) such that one of them leads to Priya's resultant and the other leads to Subhangi's resultant.

4. **Checking the Options**:
- We will check each option to see if we can derive the correct resultant vectors.

**Option A**: \( \vec{A} = 3 \hat{i} - 2 \hat{j} \) and \( \vec{B} = -2 \hat{i} + 2 \hat{j} \)
- Resultant:
\[ \vec{R} = (3 - 2) \hat{i} + (-2 + 2) \hat{j} = 1 \hat{i} + 0 \hat{j} = \hat{i} \quad \text{(Incorrect)} \]

**Option B**: \( \vec{A} = -3 \hat{i} + 2 \hat{j} \) and \( \vec{B} = 2 \hat{i} - 2 \hat{j} \)
- Resultant:
\[ \vec{R} = (-3 + 2) \hat{i} + (2 - 2) \hat{j} = -1 \hat{i} + 0 \hat{j} = -\hat{i} \quad \text{(Incorrect)} \]

**Option C**: \( \vec{A} = 3 \hat{i} - 2 \hat{j} \) and \( \vec{B} = -2 \hat{i} + 2 \hat{j} \)
- Resultant:
\[ \vec{R} = (3 - 2) \hat{i} + (-2 + 2) \hat{j} = 1 \hat{i} + 0 \hat{j} = \hat{i} \quad \text{(Incorrect)} \]

**Option D**: \( \vec{A} = 3 \hat{i} + 2 \hat{j} \) and \( \vec{B} = -2 \hat{i} + 2 \hat{j} \)
- Resultant:
\[ \vec{R} = (3 - 2) \hat{i} + (2 + 2) \hat{j} = 1 \hat{i} + 4 \hat{j} = \hat{i} + 4 \hat{j} \quad \text{(Correct)} \]

5. **Conclusion**:
- The correct option that contains the original two vectors is **Option D**.