If `a+b+c=0" and "abs(a)=3, abs(b)=5" and "abs(c)=7` then the angle between a and b is

Options

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

Answer

To find the angle between vectors \( a \) and \( b \) given the conditions \( a + b + c = 0 \), \( |a| = 3 \), \( |b| = 5 \), and \( |c| = 7 \), we can follow these steps:

### Step 1: Express \( c \) in terms of \( a \) and \( b \)
From the equation \( a + b + c = 0 \), we can express \( c \) as:
\[ c = - (a + b) \]

### Step 2: Use the magnitudes of the vectors
We know the magnitudes of the vectors:
\[ |a|^2 = 3^2 = 9, \quad |b|^2 = 5^2 = 25, \quad |c|^2 = 7^2 = 49 \]

### Step 3: Apply the magnitude condition
Using the expression for \( c \):
\[ |c|^2 = |-(a + b)|^2 = |a + b|^2 \]
Thus, we have:
\[ |a + b|^2 = |a|^2 + |b|^2 + 2(a \cdot b) \]
Substituting the known values:
\[ 49 = 9 + 25 + 2(a \cdot b) \]
This simplifies to:
\[ 49 = 34 + 2(a \cdot b) \]
\[ 2(a \cdot b) = 49 - 34 = 15 \]
\[ a \cdot b = \frac{15}{2} = 7.5 \]

### Step 4: Relate the dot product to the angle
The dot product \( a \cdot b \) can also be expressed in terms of the magnitudes and the angle \( \theta \) between them:
\[ a \cdot b = |a| |b| \cos \theta \]
Substituting the magnitudes:
\[ 7.5 = 3 \cdot 5 \cos \theta \]
\[ 7.5 = 15 \cos \theta \]
\[ \cos \theta = \frac{7.5}{15} = \frac{1}{2} \]

### Step 5: Find the angle \( \theta \)
The angle \( \theta \) for which \( \cos \theta = \frac{1}{2} \) is:
\[ \theta = \frac{\pi}{3} \quad \text{(or } 60^\circ\text{)} \]

### Conclusion
Thus, the angle between vectors \( a \) and \( b \) is:
\[ \theta = \frac{\pi}{3} \]

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