Choose the correct option:
The two vectors `vec(A)` and `vec(B)` are drawn from a common point and `vec(C)=vec(A)+vec(B)` then angle between `vec(A)` and `vec(B)` is

Options

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

Answer

To solve the problem, we need to find the angle between two vectors \(\vec{A}\) and \(\vec{B}\) given that \(\vec{C} = \vec{A} + \vec{B}\).

### Step-by-Step Solution:

1. **Understanding the Vectors**:
We have two vectors \(\vec{A}\) and \(\vec{B}\) originating from a common point. The resultant vector \(\vec{C}\) is the vector sum of \(\vec{A}\) and \(\vec{B}\).

2. **Using the Law of Cosines**:
The magnitude of the resultant vector \(\vec{C}\) can be expressed using the law of cosines:
\[ |\vec{C}|^2 = |\vec{A}|^2 + |\vec{B}|^2 + 2 |\vec{A}| |\vec{B}| \cos(\theta) \]
where \(\theta\) is the angle between vectors \(\vec{A}\) and \(\vec{B}\).

3. **Setting Up the Equation**:
Since \(\vec{C} = \vec{A} + \vec{B}\), we can denote:
\[ |\vec{C}|^2 = |\vec{A}|^2 + |\vec{B}|^2 + 2 |\vec{A}| |\vec{B}| \cos(\theta) \]

4. **Analyzing the Cases**:
- If \(\theta = 90^\circ\), then \(\cos(90^\circ) = 0\). This simplifies our equation to:
\[ |\vec{C}|^2 = |\vec{A}|^2 + |\vec{B}|^2 \]
which is true for perpendicular vectors.
- If \(\theta < 90^\circ\), then \(\cos(\theta) > 0\), meaning that \(|\vec{C}|^2 > |\vec{A}|^2 + |\vec{B}|^2\).
- If \(\theta > 90^\circ\), then \(\cos(\theta) < 0\), leading to \(|\vec{C}|^2 < |\vec{A}|^2 + |\vec{B}|^2\).

5. **Conclusion**:
The angle \(\theta\) between the vectors \(\vec{A}\) and \(\vec{B}\) can be determined as follows:
- If \(\theta = 90^\circ\), then \(|\vec{C}|^2 = |\vec{A}|^2 + |\vec{B}|^2\).
- If \(\theta < 90^\circ\), then \(|\vec{C}|^2 > |\vec{A}|^2 + |\vec{B}|^2\).
- If \(\theta > 90^\circ\), then \(|\vec{C}|^2 < |\vec{A}|^2 + |\vec{B}|^2\).

Thus, the angle between \(\vec{A}\) and \(\vec{B}\) is \(90^\circ\) when \(\vec{C} = \vec{A} + \vec{B}\).

### Final Answer:
The angle between \(\vec{A}\) and \(\vec{B}\) is \(90^\circ\).