A vector has a magnitude x . If it is rotated by an angle `theta` then magnitude of change in vector is nx. Match the following two columns.
`{:(,"Column I ",,"Column II"),((A),theta=60^(@),(P),n=sqrt(3)),((B),theta=90^(@),(Q),n=1),((C), theta=120^(@),(r),n=sqrt(2)),((D), theta=180^(@),(s),n=2):}`

Explanation

To solve the problem, we need to determine the change in magnitude of a vector when it is rotated by a given angle \( \theta \). The change in magnitude is given as \( n \cdot x \), where \( x \) is the original magnitude of the vector.

### Step-by-Step Solution:

1. **Understanding the Change in Vector Magnitude**:
The change in magnitude of the vector when rotated by an angle \( \theta \) can be expressed using the formula:
\[ \Delta r = \sqrt{r_1^2 + r_2^2 - 2r_1r_2 \cos(\theta)} \]
where \( r_1 = x \) and \( r_2 = x \) (since both vectors have the same magnitude).

2. **Simplifying the Expression**:
Substituting \( r_1 \) and \( r_2 \) into the equation:
\[ \Delta r = \sqrt{x^2 + x^2 - 2x^2 \cos(\theta)} = \sqrt{2x^2(1 - \cos(\theta))} \]
This simplifies to:
\[ \Delta r = x \sqrt{2(1 - \cos(\theta))} \]

3. **Using the Sine Double Angle Identity**:
We can use the identity \( 1 - \cos(\theta) = 2\sin^2(\theta/2) \):
\[ \Delta r = x \sqrt{2 \cdot 2\sin^2(\theta/2)} = 2x \sin(\theta/2) \]

4. **Finding \( n \)**:
We know that \( \Delta r = n \cdot x \), thus:
\[ n = \frac{\Delta r}{x} = 2 \sin(\theta/2) \]

5. **Calculating \( n \) for Different Angles**:
Now we will calculate \( n \) for each angle given in the options:

- **For \( \theta = 60^\circ \)**:
\[ n = 2 \sin(30^\circ) = 2 \cdot \frac{1}{2} = 1 \quad \text{(matches with Q)} \]

- **For \( \theta = 90^\circ \)**:
\[ n = 2 \sin(45^\circ) = 2 \cdot \frac{1}{\sqrt{2}} = \sqrt{2} \quad \text{(matches with R)} \]

- **For \( \theta = 120^\circ \)**:
\[ n = 2 \sin(60^\circ) = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \quad \text{(matches with P)} \]

- **For \( \theta = 180^\circ \)**:
\[ n = 2 \sin(90^\circ) = 2 \cdot 1 = 2 \quad \text{(matches with S)} \]

### Final Matching:
- A (60°) matches with Q (n = 1)
- B (90°) matches with R (n = √2)
- C (120°) matches with P (n = √3)
- D (180°) matches with S (n = 2)

### Summary of Matches:
- A - Q
- B - R
- C - P
- D - S