Assertion (A) : The position of a particle on a rectangular coordinate system is (3,2,5). Then its position vector be `2hat(i)+5hat(j)+3hat(k)`.
Reason (R ) : The displacement vector of the particle that moves from point P(2,3,5) to point Q(3,4,5) is `hat(i)+hat(j)`.

Options

A. Both A and R are true and R is NOT the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false
D. A is false but R is True

Answer

To solve the problem, we need to analyze both the assertion (A) and the reason (R) provided in the question.

### Step 1: Analyze the Assertion (A)

The assertion states that the position of a particle on a rectangular coordinate system is (3, 2, 5). It claims that the position vector is given by \(2\hat{i} + 5\hat{j} + 3\hat{k}\).

**Solution:**
1. The position vector of a point in a 3D coordinate system (x, y, z) is represented as \(x\hat{i} + y\hat{j} + z\hat{k}\).
2. For the point (3, 2, 5):
- \(x = 3\), \(y = 2\), \(z = 5\)
- Therefore, the position vector is \(3\hat{i} + 2\hat{j} + 5\hat{k}\).
3. The assertion claims the position vector is \(2\hat{i} + 5\hat{j} + 3\hat{k}\), which is incorrect.

**Conclusion for Assertion (A):** The assertion is **false**.

### Step 2: Analyze the Reason (R)

The reason states that the displacement vector of the particle moving from point \(P(2, 3, 5)\) to point \(Q(3, 4, 5)\) is given by \(\hat{i} + \hat{j}\).

**Solution:**
1. The position vector of point \(P\) is \(2\hat{i} + 3\hat{j} + 5\hat{k}\).
2. The position vector of point \(Q\) is \(3\hat{i} + 4\hat{j} + 5\hat{k}\).
3. The displacement vector \(PQ\) is calculated as:
\[ PQ = Q - P = (3\hat{i} + 4\hat{j} + 5\hat{k}) - (2\hat{i} + 3\hat{j} + 5\hat{k}) \]
4. Performing the subtraction:
- For \(\hat{i}\): \(3 - 2 = 1\)
- For \(\hat{j}\): \(4 - 3 = 1\)
- For \(\hat{k}\): \(5 - 5 = 0\)
5. Therefore, the displacement vector \(PQ\) is:
\[ PQ = 1\hat{i} + 1\hat{j} + 0\hat{k} = \hat{i} + \hat{j} \]

**Conclusion for Reason (R):** The reason is **true**.

### Final Conclusion
- Assertion (A) is false.
- Reason (R) is true.