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 `a.b +b.c +c.a` equal to

Options

A. `-83`
B. `-(83)/(2)`
C. 75
D. `-(75)/(2)`

Answer

To solve the problem, we need to find the value of \( a \cdot b + b \cdot c + c \cdot a \) given that \( a + b + c = 0 \) and the magnitudes of the vectors are \( |a| = 3 \), \( |b| = 5 \), and \( |c| = 7 \).

### Step-by-step Solution:

1. **Understanding the Relationship**:
Since \( a + b + c = 0 \), we can express \( c \) as:
\[ c = - (a + b) \]

2. **Using the Magnitudes**:
The magnitudes of the vectors are given as:
\[ |a|^2 = 3^2 = 9, \quad |b|^2 = 5^2 = 25, \quad |c|^2 = 7^2 = 49 \]

3. **Using the Dot Product Formula**:
We know that:
\[ |a + b + c|^2 = |0|^2 = 0 \]
Expanding \( |a + b + c|^2 \) gives us:
\[ |a|^2 + |b|^2 + |c|^2 + 2(a \cdot b + b \cdot c + c \cdot a) = 0 \]

4. **Substituting the Values**:
Substituting the values of the magnitudes:
\[ 9 + 25 + 49 + 2(a \cdot b + b \cdot c + c \cdot a) = 0 \]
Simplifying this:
\[ 83 + 2(a \cdot b + b \cdot c + c \cdot a) = 0 \]

5. **Solving for the Dot Products**:
Rearranging the equation:
\[ 2(a \cdot b + b \cdot c + c \cdot a) = -83 \]
Dividing by 2:
\[ a \cdot b + b \cdot c + c \cdot a = -\frac{83}{2} \]

### Final Answer:
Thus, the value of \( a \cdot b + b \cdot c + c \cdot a \) is:
\[ \boxed{-\frac{83}{2}} \]