If `veca+vecb+vecc=vec0, |veca| = 3, |vecb| = 5, |vecc| = 7`, then angle between `veca` and `vecb` is : a. `(pi)/(2)` b. `(pi)/(3)` c. `(pi)/4` d. `(pi)/(6)`

Options

A. `(pi)/(2)`
B. `(pi)/(3)`
C. `(pi)/4`
D. `(pi)/(6)`

Explanation

To solve the problem, we start with the equation given in the question:

1. **Given Equation**:
\[ \vec{a} + \vec{b} + \vec{c} = \vec{0} \]
Rearranging gives:
\[ \vec{a} + \vec{b} = -\vec{c} \]

2. **Taking Magnitudes**:
Taking the magnitudes on both sides:
\[ |\vec{a} + \vec{b}| = |\vec{c}| \]
Squaring both sides:
\[ |\vec{a} + \vec{b}|^2 = |\vec{c}|^2 \]

3. **Expanding the Left Side**:
Using the formula for the magnitude of a sum of vectors:
\[ |\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2 \vec{a} \cdot \vec{b} \]
Thus, we have:
\[ |\vec{a}|^2 + |\vec{b}|^2 + 2 \vec{a} \cdot \vec{b} = |\vec{c}|^2 \]

4. **Substituting Known Values**:
We know:
\[ |\vec{a}| = 3, \quad |\vec{b}| = 5, \quad |\vec{c}| = 7 \]
Substituting these values in:
\[ 3^2 + 5^2 + 2 \vec{a} \cdot \vec{b} = 7^2 \]
This simplifies to:
\[ 9 + 25 + 2 \vec{a} \cdot \vec{b} = 49 \]

5. **Simplifying the Equation**:
Combining the constants:
\[ 34 + 2 \vec{a} \cdot \vec{b} = 49 \]
Rearranging gives:
\[ 2 \vec{a} \cdot \vec{b} = 49 - 34 = 15 \]

6. **Using the Dot Product**:
The dot product can be expressed as:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]
Therefore:
\[ 2 |\vec{a}| |\vec{b}| \cos \theta = 15 \]
Substituting the magnitudes:
\[ 2 \cdot 3 \cdot 5 \cos \theta = 15 \]

7. **Solving for Cosine**:
This simplifies to:
\[ 30 \cos \theta = 15 \]
Dividing both sides by 30:
\[ \cos \theta = \frac{15}{30} = \frac{1}{2} \]

8. **Finding the Angle**:
The angle whose cosine is \( \frac{1}{2} \) is:
\[ \theta = \frac{\pi}{3} \]

Thus, the angle between \( \vec{a} \) and \( \vec{b} \) is \( \frac{\pi}{3} \).

### Final Answer:
The angle between \( \vec{a} \) and \( \vec{b} \) is \( \frac{\pi}{3} \).