If `veca +vecb +vecc=0, |veca|=3,|vecb|=5, |vecc|=7` , then find the angle between `veca and vecb`.

Options

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

Explanation

To solve the problem, we start with the given vector equation and magnitudes:

1. **Given Information**:
- \( \vec{a} + \vec{b} + \vec{c} = 0 \)
- \( |\vec{a}| = 3 \)
- \( |\vec{b}| = 5 \)
- \( |\vec{c}| = 7 \)

2. **Rearranging the Equation**:
We can rearrange the vector equation to isolate one of the vectors:
\[ \vec{c} = -(\vec{a} + \vec{b}) \]

3. **Applying Magnitudes**:
Taking the magnitude on both sides:
\[ |\vec{c}| = |-(\vec{a} + \vec{b})| = |\vec{a} + \vec{b}| \]
Since the magnitude of a vector is always positive, we can write:
\[ 7 = |\vec{a} + \vec{b}| \]

4. **Using the Magnitude Formula**:
The magnitude of the sum of two vectors can be expressed using the formula:
\[ |\vec{a} + \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2 + 2 |\vec{a}| |\vec{b}| \cos \theta} \]
where \( \theta \) is the angle between vectors \( \vec{a} \) and \( \vec{b} \).

5. **Substituting Known Values**:
Substituting the known magnitudes into the formula:
\[ 7 = \sqrt{3^2 + 5^2 + 2 \cdot 3 \cdot 5 \cdot \cos \theta} \]
This simplifies to:
\[ 7 = \sqrt{9 + 25 + 30 \cos \theta} \]
\[ 7 = \sqrt{34 + 30 \cos \theta} \]

6. **Squaring Both Sides**:
To eliminate the square root, we square both sides:
\[ 49 = 34 + 30 \cos \theta \]

7. **Rearranging the Equation**:
Rearranging gives:
\[ 30 \cos \theta = 49 - 34 \]
\[ 30 \cos \theta = 15 \]

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

9. **Finding the Angle**:
The angle \( \theta \) whose cosine is \( \frac{1}{2} \) is:
\[ \theta = 60^\circ \]

Thus, the angle between \( \vec{a} \) and \( \vec{b} \) is \( 60^\circ \).