Read the following information carefully and answer the questions given below
`a+b+c= 0` such that `|a|= 3, |b|=5 and |c|=7`
What is the angle between a and b?

Options

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

Answer

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

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

### Step 2: Use the property of magnitudes
Taking the magnitude of both sides, we have:
\[ |c| = |-(a + b)| = |a + b| \]
Since \( |c| = 7 \), we can write:
\[ |a + b| = 7 \]

### Step 3: Apply the formula for the magnitude of the sum of two vectors
The magnitude of the sum of two vectors can be expressed using the cosine of the angle \( \theta \) between them:
\[ |a + b|^2 = |a|^2 + |b|^2 + 2|a||b|\cos(\theta) \]
Substituting the known values:
\[ 7^2 = 3^2 + 5^2 + 2 \cdot 3 \cdot 5 \cdot \cos(\theta) \]

### Step 4: Calculate the squares
Calculating the squares:
\[ 49 = 9 + 25 + 30 \cos(\theta) \]
Simplifying the left side:
\[ 49 = 34 + 30 \cos(\theta) \]

### Step 5: Isolate the cosine term
Rearranging the equation gives:
\[ 49 - 34 = 30 \cos(\theta) \]
\[ 15 = 30 \cos(\theta) \]

### Step 6: Solve for \( \cos(\theta) \)
Dividing both sides by 30:
\[ \cos(\theta) = \frac{15}{30} = \frac{1}{2} \]

### Step 7: Find the angle \( \theta \)
The angle \( \theta \) whose cosine is \( \frac{1}{2} \) is:
\[ \theta = 60^\circ \]

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

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