The magnitude of vector `vec(A),vec(B)` and `vec(C )` are respectively 12,5 and 13 unit and `vec(A)+vec(B)= vec(C )` then the angle between `vec(A)` and `vec(B)` is

Options

A. 0
B. `pi`
C. `pi//2`
D. `pi//4`

Explanation

To find the angle between the vectors \(\vec{A}\) and \(\vec{B}\), we can use the information given in the problem along with the properties of vectors.

### Step-by-Step Solution:

1. **Identify the Magnitudes**:
- The magnitudes of the vectors are given as follows:
- \(|\vec{A}| = 12\)
- \(|\vec{B}| = 5\)
- \(|\vec{C}| = 13\)

2. **Use the Vector Addition Formula**:
- According to the problem, we have:
\[ \vec{A} + \vec{B} = \vec{C} \]

3. **Apply the Law of Cosines**:
- The Law of Cosines states that for any two vectors \(\vec{A}\) and \(\vec{B}\), the magnitude of their resultant vector \(\vec{C}\) can be expressed as:
\[ |\vec{C}|^2 = |\vec{A}|^2 + |\vec{B}|^2 + 2 |\vec{A}| |\vec{B}| \cos \theta \]
- Here, \(\theta\) is the angle between \(\vec{A}\) and \(\vec{B}\).

4. **Substitute the Known Values**:
- Plugging in the magnitudes:
\[ 13^2 = 12^2 + 5^2 + 2 \cdot 12 \cdot 5 \cdot \cos \theta \]
- This simplifies to:
\[ 169 = 144 + 25 + 120 \cos \theta \]

5. **Simplify the Equation**:
- Combine the terms on the right side:
\[ 169 = 169 + 120 \cos \theta \]
- Rearranging gives:
\[ 0 = 120 \cos \theta \]

6. **Solve for \(\cos \theta\)**:
- This implies:
\[ \cos \theta = 0 \]

7. **Determine the Angle**:
- The angle \(\theta\) for which \(\cos \theta = 0\) is:
\[ \theta = 90^\circ \]

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