A vector `vec(a)` is turned without a change in its length through a small angle `d theta`. Find the value of `|Deltavec(a)|`

Options

A. `0`
B. `a.d theta`
C. `0`
D. None of these

Explanation

To solve the problem, we need to determine the change in a vector \( \vec{a} \) when it is rotated through a small angle \( d\theta \) without changing its length. Let's break down the steps to find \( |\Delta \vec{a}| \).

### Step-by-Step Solution:

1. **Understanding the Vector Rotation**:
- When a vector \( \vec{a} \) is rotated through a small angle \( d\theta \), it maintains its length. This means that the magnitude of the vector remains constant.

2. **Visualizing the Rotation**:
- Imagine the vector \( \vec{a} \) originating from a point and pointing in a certain direction. After rotating it by an angle \( d\theta \), the new position of the vector will also have the same length but will point in a slightly different direction.

3. **Identifying the Arc Length**:
- The rotation of the vector can be visualized as moving along the arc of a circle where the radius is equal to the magnitude of the vector \( |\vec{a}| \). The length of the arc corresponding to the angle \( d\theta \) can be expressed as:
\[ \text{Arc Length} = |\Delta \vec{a}| = r \cdot d\theta \]
- Here, \( r \) is the radius of the circle, which is equal to \( |\vec{a}| \).

4. **Substituting the Radius**:
- Since the radius \( r \) is equal to the magnitude of the vector \( |\vec{a}| \), we can substitute this into the equation:
\[ |\Delta \vec{a}| = |\vec{a}| \cdot d\theta \]

5. **Final Expression**:
- Therefore, the change in the vector \( \vec{a} \) due to the rotation through the small angle \( d\theta \) is given by:
\[ |\Delta \vec{a}| = |\vec{a}| \cdot d\theta \]

### Conclusion:
The value of \( |\Delta \vec{a}| \) is \( |\vec{a}| \cdot d\theta \).