Two vectors `vecA and vecB` inclined at an angle `theta` have a resultant `vecR` which makes an angle `alpha` with `vecA` and angle `beta` with `vecB`. Let the magnitudes of the vectors `vecA, vecB and vecR` be represented by A, B and R respectively. Which of the following relations is not correct ?

Options

A. `(R)/(sin(alpha+beta))=(A)/(sin alpha) = (B)/(sin beta)`
B. `R sin alpha = B sin (alpha + beta)`
C. `A sin alpha = B sin beta`
D. `R sin beta = A sin (alpha + beta)`

Explanation

To solve the problem regarding the two vectors \(\vec{A}\) and \(\vec{B}\) inclined at an angle \(\theta\) and their resultant \(\vec{R}\), we will analyze the relationships between the angles and magnitudes of these vectors. The goal is to identify which of the given relations is not correct.

### Step-by-Step Solution:

1. **Understanding the Vectors and Angles**:
- Let \(\vec{A}\) and \(\vec{B}\) be two vectors with magnitudes \(A\) and \(B\) respectively.
- The angle between \(\vec{A}\) and \(\vec{B}\) is \(\theta\).
- The resultant vector \(\vec{R}\) makes an angle \(\alpha\) with \(\vec{A}\) and an angle \(\beta\) with \(\vec{B}\).

2. **Drawing the Diagram**:
- Draw vector \(\vec{A}\) and vector \(\vec{B}\) such that they form an angle \(\theta\).
- The resultant vector \(\vec{R}\) can be represented as the diagonal of a parallelogram formed by \(\vec{A}\) and \(\vec{B}\).

3. **Using the Law of Sines**:
- From the triangle formed by the vectors, we can apply the Law of Sines:
\[ \frac{R}{\sin(\theta)} = \frac{A}{\sin(\beta)} = \frac{B}{\sin(\alpha)} \]
- This gives us three relationships based on the angles and magnitudes of the vectors.

4. **Finding Relationships**:
- From the Law of Sines, we can derive:
\[ R \sin(\beta) = A \sin(\theta) \]
\[ R \sin(\alpha) = B \sin(\theta) \]

5. **Combining Relationships**:
- We can combine the relationships to find:
\[ \frac{R}{\sin(\alpha + \beta)} = \frac{A}{\sin(\beta)} = \frac{B}{\sin(\alpha)} \]
- This leads to the conclusion that:
\[ A \sin(\alpha) = B \sin(\beta) \]

6. **Identifying the Incorrect Relation**:
- The relations derived from the Law of Sines are valid, but we need to check which of the provided options does not hold true.
- The incorrect relation will be one that does not satisfy the derived equations.

### Conclusion:
After analyzing the relationships, the incorrect option among the provided choices is the one that does not hold true based on the derived equations from the Law of Sines.