The resultant of ` vec(a)` and ` vec(b)` makes `alpha` with `vec(a)` and `beta` with `vec(b)`, then (a,b represent magnitudes of respective vectors) :

Options

A. `alpha lt beta`
B. `alpha lt beta if a lt b`
C. `alpha lt beta if a gt b`
D. `alpha lt beta if a =b`

Answer

To solve the problem, we need to analyze the relationship between the angles α and β formed by the resultant vector with vectors A and B, respectively. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
We have two vectors, A and B, with magnitudes \( a \) and \( b \). The resultant vector \( R \) makes an angle \( \alpha \) with vector \( A \) and an angle \( \beta \) with vector \( B \).

### Step 2: Apply the Law of Sines
According to the law of sines in the context of vector addition, we can relate the angles and the magnitudes of the vectors:
\[ \frac{a}{\sin(\beta)} = \frac{b}{\sin(\alpha)} = \frac{R}{\sin(\theta)} \]
where \( \theta \) is the angle between vectors A and B.

### Step 3: Analyze the Angles
From the above relationship, we can derive:
\[ \frac{a}{b} = \frac{\sin(\beta)}{\sin(\alpha)} \]
This implies that the ratio of the magnitudes of the vectors is equal to the ratio of the sines of the angles they make with the resultant.

### Step 4: Determine the Relationship Between α and β
- If \( a > b \), then \( \frac{a}{b} > 1 \), which implies \( \sin(\beta) > \sin(\alpha) \). Since the sine function is increasing in the range \( 0^\circ \) to \( 90^\circ \), this means \( \beta > \alpha \).
- Conversely, if \( b > a \), then \( \frac{a}{b} < 1 \), which implies \( \sin(\beta) < \sin(\alpha) \), leading to \( \beta < \alpha \).

### Step 5: Conclusion
The relationship between the angles α and β depends on the magnitudes of the vectors A and B:
- If \( a > b \), then \( \beta > \alpha \).
- If \( b > a \), then \( \alpha > \beta \).
- If \( a = b \), then \( \alpha = \beta \).

### Final Answer
Thus, we can conclude that the angles α and β are related to the magnitudes of the vectors A and B. The correct statement is:
- If \( a > b \), then \( \beta > \alpha \) (Option C is correct).