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. direction
B. magnitude
C. direction as well as magnitude
D. none of these

Explanation

To solve the problem, we need to analyze the situation involving two vectors \( \vec{A} \) and \( \vec{B} \) that are inclined at an angle \( \theta \). We will determine what happens to the resultant vector \( \vec{R} \) when the directions of \( \vec{A} \) and \( \vec{B} \) are interchanged.

### Step-by-Step Solution:

1. **Understanding the Vectors**:
- Let \( \vec{A} \) and \( \vec{B} \) be two vectors inclined at an angle \( \theta \).
- The resultant vector \( \vec{R} \) can be found using the law of cosines.

2. **Finding the Magnitude of the Resultant**:
- The magnitude of the resultant vector \( \vec{R} \) when \( \vec{A} \) and \( \vec{B} \) are at an angle \( \theta \) is given by:
\[ R = \sqrt{A^2 + B^2 + 2AB \cos(\theta)} \]

3. **Finding the Direction of the Resultant**:
- The angle \( \alpha \) that the resultant \( \vec{R} \) makes with \( \vec{A} \) can be found using the tangent function:
\[ \tan(\alpha) = \frac{B \sin(\theta)}{A + B \cos(\theta)} \]

4. **Interchanging the Directions of Vectors**:
- When the directions of \( \vec{A} \) and \( \vec{B} \) are interchanged, \( \vec{B} \) will now be the first vector and \( \vec{A} \) will be the second vector.
- The resultant in this case can still be calculated using the same formula for magnitude:
\[ R' = \sqrt{B^2 + A^2 + 2AB \cos(\theta)} \]
- The magnitude remains the same since addition is commutative.

5. **Finding the New Direction of the Resultant**:
- The new angle \( \alpha' \) that the resultant \( \vec{R'} \) makes with \( \vec{B} \) can be calculated as:
\[ \tan(\alpha') = \frac{A \sin(\theta)}{B + A \cos(\theta)} \]

6. **Conclusion**:
- The magnitude of the resultant remains the same, but the direction changes due to the interchange of the vectors.
- Therefore, the answer to the question is that the magnitude remains the same, but the direction changes.

### Final Answer:
- The resultant will have the same **magnitude** but a different **direction**.