If ` veca +vecb +vecc =vec0, |veca| =3 , |vecb|=5 and |vecc| =7` , then the angle between ` veca and vecb` is

Options

A. `(pi)/(6)`
B. `(2pi)/(3)`
C. `(5pi)/(3)`
D. `(pi)/(3)`

Answer

To solve the problem, we need to find the angle between the vectors \( \vec{a} \) and \( \vec{b} \) given the equation \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \) and the magnitudes \( |\vec{a}| = 3 \), \( |\vec{b}| = 5 \), and \( |\vec{c}| = 7 \).

### Step-by-Step Solution:

1. **Rearranging the Equation**:
\[ \vec{a} + \vec{b} = -\vec{c} \]
This means that the vector sum of \( \vec{a} \) and \( \vec{b} \) is equal to the negative of \( \vec{c} \).

**Hint**: Remember that the negative of a vector points in the opposite direction.

2. **Taking Magnitudes**:
Taking the magnitude of both sides, we have:
\[ |\vec{a} + \vec{b}| = |\vec{c}| \]
Since \( |\vec{c}| = 7 \), we can write:
\[ |\vec{a} + \vec{b}| = 7 \]

**Hint**: The magnitude of a vector can be calculated using the Pythagorean theorem if the vectors are perpendicular.

3. **Using the Formula for Magnitude**:
The magnitude of the sum of two vectors can be expressed as:
\[ |\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2 \vec{a} \cdot \vec{b} \]
Substituting the known values:
\[ 7^2 = 3^2 + 5^2 + 2 \vec{a} \cdot \vec{b} \]
This simplifies to:
\[ 49 = 9 + 25 + 2 \vec{a} \cdot \vec{b} \]

**Hint**: Squaring the magnitudes helps eliminate the square root when calculating the magnitude of the sum.

4. **Calculating the Dot Product**:
Now, calculate:
\[ 49 = 34 + 2 \vec{a} \cdot \vec{b} \]
Rearranging gives:
\[ 2 \vec{a} \cdot \vec{b} = 49 - 34 = 15 \]
Thus,
\[ \vec{a} \cdot \vec{b} = \frac{15}{2} = 7.5 \]

**Hint**: The dot product can also be expressed in terms of the angle between the vectors.

5. **Using the Dot Product Formula**:
The dot product can also be expressed as:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]
Substituting the magnitudes:
\[ 7.5 = 3 \cdot 5 \cdot \cos \theta \]
This simplifies to:
\[ 7.5 = 15 \cos \theta \]
Thus,
\[ \cos \theta = \frac{7.5}{15} = \frac{1}{2} \]

**Hint**: The cosine of an angle gives you a way to find the angle itself.

6. **Finding the Angle**:
From \( \cos \theta = \frac{1}{2} \), we find:
\[ \theta = \cos^{-1}\left(\frac{1}{2}\right) \]
This gives:
\[ \theta = 60^\circ \]

**Hint**: Remember the common angles and their cosines; \( \cos 60^\circ = \frac{1}{2} \).

### Final Answer:
The angle between \( \vec{a} \) and \( \vec{b} \) is \( 60^\circ \) or \( \frac{\pi}{3} \) radians.