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 involving two vectors \(\vec{A}\) and \(\vec{B}\) inclined at an angle \(\theta\), and their resultant \(\vec{R}\) making an angle \(\alpha\) with \(\vec{A}\), we can follow these steps:

### Step 1: Understand the Vectors and Their Resultant
We have two vectors \(\vec{A}\) and \(\vec{B}\) that are inclined at an angle \(\theta\). The resultant vector \(\vec{R}\) can be calculated using the law of cosines and the law of sines.

### Step 2: Calculate the Magnitude of the Resultant
The magnitude of the resultant vector \(\vec{R}\) can be calculated using the formula:
\[ R = \sqrt{A^2 + B^2 + 2AB \cos(\theta)} \]
where \(A\) and \(B\) are the magnitudes of vectors \(\vec{A}\) and \(\vec{B}\), respectively.

### Step 3: Determine the Angle \(\alpha\)
The angle \(\alpha\) that the resultant \(\vec{R}\) makes with vector \(\vec{A}\) can be found using the sine rule:
\[ \frac{R}{\sin(\alpha)} = \frac{B}{\sin(\theta)} \]
From this, we can rearrange to find \(\alpha\):
\[ \sin(\alpha) = \frac{R \cdot \sin(\theta)}{B} \]

### Step 4: Interchange the Directions of \(\vec{A}\) and \(\vec{B}\)
When the directions of \(\vec{A}\) and \(\vec{B}\) are interchanged, we denote the new vectors as \(\vec{B'}\) and \(\vec{A'}\). The resultant vector will still have the same magnitude \(R\) but will make a different angle, say \(\alpha'\), with the new vector \(\vec{B'}\).

### Step 5: Analyze the Resultant After Interchanging
The resultant vector's magnitude remains the same, but the angle \(\alpha'\) can be calculated similarly using the law of sines:
\[ \sin(\alpha') = \frac{R \cdot \sin(\theta)}{A} \]
This shows that the resultant vector's direction changes, but the magnitude remains constant.

### Conclusion
Thus, the resultant vector \(\vec{R}\) retains the same magnitude regardless of whether the vectors \(\vec{A}\) and \(\vec{B}\) are interchanged, but the angles \(\alpha\) and \(\alpha'\) will differ.