The resultant of `vecA and vecB` makes an angle `alpha` with `vecA` and `beta and vecB`,

Options

A. `alphaltbeta`
B. `alphaltbeta if A ltB`
C. `alphaltbeta if AgtB`
D. `alphaltbeta if A=B.`

Answer

To solve the problem, we need to analyze the relationship between the angles \( \alpha \) and \( \beta \) formed by the resultant vector \( \vec{R} \) with vectors \( \vec{A} \) and \( \vec{B} \) respectively.

### Step-by-Step Solution:

1. **Understand the Vectors and Resultant**:
- Let \( \vec{A} \) and \( \vec{B} \) be two vectors.
- The resultant vector \( \vec{R} \) can be represented as \( \vec{R} = \vec{A} + \vec{B} \).
- The angles \( \alpha \) and \( \beta \) are defined as the angles between \( \vec{R} \) and \( \vec{A} \), and \( \vec{R} \) and \( \vec{B} \) respectively.

2. **Effect of Magnitude Changes**:
- When the magnitude of \( \vec{A} \) increases, the resultant \( \vec{R} \) shifts closer to \( \vec{A} \).
- This means that the angle \( \alpha \) (between \( \vec{R} \) and \( \vec{A} \)) decreases.
- Conversely, as \( \vec{R} \) shifts closer to \( \vec{A} \), the angle \( \beta \) (between \( \vec{R} \) and \( \vec{B} \)) increases.

3. **Analyzing Different Cases**:
- If \( |\vec{A}| > |\vec{B}| \):
- \( \beta \) will be greater than \( \alpha \) since \( \vec{R} \) is closer to \( \vec{A} \).
- If \( |\vec{B}| > |\vec{A}| \):
- \( \alpha \) will be greater than \( \beta \) since \( \vec{R} \) is closer to \( \vec{B} \).
- If \( |\vec{A}| = |\vec{B}| \):
- Then \( \alpha = \beta \).

4. **Conclusion**:
- The relationship between \( \alpha \) and \( \beta \) depends on the magnitudes of \( \vec{A} \) and \( \vec{B} \):
- If \( |\vec{A}| > |\vec{B}| \), then \( \beta > \alpha \).
- If \( |\vec{B}| > |\vec{A}| \), then \( \alpha > \beta \).
- If \( |\vec{A}| = |\vec{B}| \), then \( \alpha = \beta \).

### Final Answer:
The relation between \( \alpha \) and \( \beta \) is:
- \( \beta > \alpha \) if \( |\vec{A}| > |\vec{B}| \)
- \( \alpha > \beta \) if \( |\vec{B}| > |\vec{A}| \)
- \( \alpha = \beta \) if \( |\vec{A}| = |\vec{B}| \)