The two vectors `vecA` and `vecB` are drawn from a common point and `vecC=vecA+vecB`. Then, the angle between `vecA` and `vecB` is

Options

A. `90^(@)` if `C^(2)gtA^(2)+B^(2)`
B. greater than `90^(@)` if `C^(2)ltA^(2)+B^(2)`
C. greater than `90^(@)` if `C^(2)gtA^(2)+B^(2)`
D. less than `90^(@)` if `C^(2)gtA^(2)+B^(2)`

Answer

To solve the problem regarding the angle between the vectors \(\vec{A}\) and \(\vec{B}\), we can use the law of vector addition, specifically the parallelogram law.

### Step-by-Step Solution:

1. **Understanding Vector Addition**:
- We have two vectors \(\vec{A}\) and \(\vec{B}\) that originate from a common point. The resultant vector \(\vec{C}\) is defined as \(\vec{C} = \vec{A} + \vec{B}\).

2. **Using the Parallelogram Law**:
- According to the parallelogram law of vector addition, the magnitude of the resultant vector \(\vec{C}\) can be expressed as:
\[ C^2 = A^2 + B^2 + 2AB \cos(\theta) \]
where \(C\) is the magnitude of \(\vec{C}\), \(A\) is the magnitude of \(\vec{A}\), \(B\) is the magnitude of \(\vec{B}\), and \(\theta\) is the angle between \(\vec{A}\) and \(\vec{B}\).

3. **Analyzing the Magnitudes**:
- We can rearrange the equation to find the relationship between \(C^2\), \(A^2\), and \(B^2\):
\[ C^2 = A^2 + B^2 + 2AB \cos(\theta) \]

4. **Considering Different Cases**:
- **Case 1**: If \(\theta < 90^\circ\) (acute angle), then \(\cos(\theta) > 0\), which implies:
\[ C^2 > A^2 + B^2 \]
- **Case 2**: If \(\theta = 90^\circ\) (right angle), then \(\cos(90^\circ) = 0\), leading to:
\[ C^2 = A^2 + B^2 \]
- **Case 3**: If \(\theta > 90^\circ\) (obtuse angle), then \(\cos(\theta) < 0\), which results in:
\[ C^2 < A^2 + B^2 \]

5. **Conclusion**:
- From the analysis, we can conclude:
- If the angle \(\theta\) is acute (less than \(90^\circ\)), then \(C^2 > A^2 + B^2\).
- If the angle \(\theta\) is obtuse (greater than \(90^\circ\)), then \(C^2 < A^2 + B^2\).
- If the angle \(\theta\) is exactly \(90^\circ\), then \(C^2 = A^2 + B^2\).

### Final Answer:
The angle between \(\vec{A}\) and \(\vec{B}\) can be:
- Acute (less than \(90^\circ\)) or
- Obtuse (greater than \(90^\circ\)).