Two vectors `vec(A) and vec(B)` inclined at an angle `theta` have a resultant `vec(R )` which makes an angle `alpha` with `vec(A)`. If the directions of `vec(A) and vec(B)` are interchanged, the resultant will have the same

Options

A. magnitude the same
B. direction reverses and megnitude is doubled
C. magnitude as well as direction
D. Neither

Explanation

To solve the problem, we need to analyze the situation with two vectors \( \vec{A} \) and \( \vec{B} \) that are inclined at an angle \( \theta \). The resultant vector \( \vec{R} \) makes an angle \( \alpha \) with \( \vec{A} \). We are tasked with determining what happens to the resultant vector if the directions of \( \vec{A} \) and \( \vec{B} \) are interchanged.

### Step-by-Step Solution:

1. **Understanding the Magnitude of the Resultant Vector**:
The magnitude of the resultant vector \( \vec{R} \) when two vectors \( \vec{A} \) and \( \vec{B} \) are inclined at an angle \( \theta \) is given by the formula:
\[ R = \sqrt{A^2 + B^2 + 2AB \cos(\theta)} \]
Here, \( A \) and \( B \) are the magnitudes of vectors \( \vec{A} \) and \( \vec{B} \), respectively.

**Hint**: Remember that the magnitude of the resultant depends on the magnitudes of the individual vectors and the cosine of the angle between them.

2. **Interchanging the Directions of Vectors**:
When we interchange the directions of \( \vec{A} \) and \( \vec{B} \), the angle \( \theta \) between them remains the same, and the magnitudes \( A \) and \( B \) also remain unchanged. Therefore, the formula for the magnitude of the resultant vector \( R \) remains the same.

**Hint**: Consider how the angle between the vectors and their magnitudes affect the resultant vector's magnitude.

3. **Analyzing the Direction of the Resultant Vector**:
The resultant vector \( \vec{R} \) is more inclined towards the vector with the greater magnitude. If \( |\vec{A}| > |\vec{B}| \), then the angle \( \alpha \) between \( \vec{A} \) and \( \vec{R} \) is less than the angle between \( \vec{B} \) and \( \vec{R} \).

When we interchange the directions of \( \vec{A} \) and \( \vec{B} \), the resultant vector will still make an angle \( \alpha \) with the new direction of \( \vec{A} \) (which was originally \( \vec{B} \)). However, since the direction of \( \vec{A} \) has changed, the resultant vector \( \vec{R} \) will also change its direction to maintain the same angle \( \alpha \).

**Hint**: Think about how the direction of the resultant vector is influenced by the directions of the original vectors.

4. **Conclusion**:
Since the magnitude of the resultant vector remains the same but its direction changes when the directions of \( \vec{A} \) and \( \vec{B} \) are interchanged, we conclude that the resultant vector will have the same magnitude but a different direction.

Therefore, the correct option is that the resultant vector will have the same magnitude but its direction will change.

### Final Answer:
The resultant vector \( \vec{R} \) will have the same magnitude but a different direction when the directions of \( \vec{A} \) and \( \vec{B} \) are interchanged.