If `a+b+c=0` and `|a|=5, |b|=3` and `|c|=7`, then angle between a and b is

Options

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

Explanation

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

### Step-by-Step Solution:

1. **Understand the relationship between the vectors:**
Since \( a + b + c = 0 \), we can express \( c \) as:
\[ c = - (a + b) \]

2. **Use the property of magnitudes:**
The magnitude of vector \( c \) can be expressed in terms of vectors \( a \) and \( b \):
\[ |c|^2 = |a + b|^2 \]

3. **Apply the formula for the magnitude of the sum of two vectors:**
The magnitude of the sum of two vectors is given by:
\[ |a + b|^2 = |a|^2 + |b|^2 + 2(a \cdot b) \]
Therefore, we can write:
\[ |c|^2 = |a|^2 + |b|^2 + 2(a \cdot b) \]

4. **Substitute the given magnitudes:**
We know:
- \( |a| = 5 \) so \( |a|^2 = 25 \)
- \( |b| = 3 \) so \( |b|^2 = 9 \)
- \( |c| = 7 \) so \( |c|^2 = 49 \)

Plugging these values into the equation gives:
\[ 49 = 25 + 9 + 2(a \cdot b) \]

5. **Simplify the equation:**
Combine the constants:
\[ 49 = 34 + 2(a \cdot b) \]
Subtract 34 from both sides:
\[ 15 = 2(a \cdot b) \]

6. **Solve for \( a \cdot b \):**
Divide both sides by 2:
\[ a \cdot b = \frac{15}{2} \]

7. **Use the dot product to find the angle:**
The dot product can also be expressed in terms of the magnitudes and the cosine of the angle \( \theta \) between them:
\[ a \cdot b = |a| |b| \cos \theta \]
Substituting the known values:
\[ \frac{15}{2} = 5 \cdot 3 \cdot \cos \theta \]
Simplifying gives:
\[ \frac{15}{2} = 15 \cos \theta \]

8. **Solve for \( \cos \theta \):**
Divide both sides by 15:
\[ \cos \theta = \frac{1}{2} \]

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

### Final Answer:
The angle between vectors \( a \) and \( b \) is \( 60^\circ \).