A vector of length `l` is turned through the angle `theta` about its tail. What is the change in the position vector of its head ?

Options

A. `l cos (theta//2)`
B. `2l sin(theta//2)`
C. `2l cos(theta//2)`
D. `l sin (theta//2)`

Explanation

To solve the problem of finding the change in the position vector of the head of a vector of length \( l \) that is turned through an angle \( \theta \) about its tail, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Initial and Final Vectors**:
- Let the initial position vector of the head be \( \mathbf{R_1} \) and the final position vector after rotation be \( \mathbf{R_2} \).
- Both vectors have the same magnitude, which is \( l \).

2. **Express the Change in Position Vector**:
- The change in the position vector \( \Delta \mathbf{R} \) is given by:
\[ \Delta \mathbf{R} = \mathbf{R_2} - \mathbf{R_1} \]

3. **Use the Law of Cosines**:
- To find the magnitude of the change in position vector, we can use the Law of Cosines:
\[ |\Delta \mathbf{R}| = |\mathbf{R_2} - \mathbf{R_1}| = \sqrt{|\mathbf{R_1}|^2 + |\mathbf{R_2}|^2 - 2 |\mathbf{R_1}| |\mathbf{R_2}| \cos(\theta)} \]
- Since \( |\mathbf{R_1}| = l \) and \( |\mathbf{R_2}| = l \), we substitute these values into the equation:
\[ |\Delta \mathbf{R}| = \sqrt{l^2 + l^2 - 2l \cdot l \cos(\theta)} \]

4. **Simplify the Expression**:
- This simplifies to:
\[ |\Delta \mathbf{R}| = \sqrt{2l^2 - 2l^2 \cos(\theta)} \]
- Factor out \( 2l^2 \):
\[ |\Delta \mathbf{R}| = \sqrt{2l^2(1 - \cos(\theta))} \]

5. **Use the Trigonometric Identity**:
- We know from trigonometric identities that:
\[ 1 - \cos(\theta) = 2 \sin^2\left(\frac{\theta}{2}\right) \]
- Substitute this into the equation:
\[ |\Delta \mathbf{R}| = \sqrt{2l^2 \cdot 2 \sin^2\left(\frac{\theta}{2}\right)} \]

6. **Final Simplification**:
- This simplifies to:
\[ |\Delta \mathbf{R}| = \sqrt{4l^2 \sin^2\left(\frac{\theta}{2}\right)} = 2l \sin\left(\frac{\theta}{2}\right) \]

### Conclusion:
Thus, the change in the position vector of the head is:
\[ \Delta \mathbf{R} = 2l \sin\left(\frac{\theta}{2}\right) \]