If `abs(a)=5, abs(b)=4, abs(c)=3` then the value of `(a*b+b*c+c*a)` given that `a+b+c=0`

Options

A. 25
B. 50
C. -25
D. -50

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 \( |a| = 5 \), \( |b| = 4 \), and \( |c| = 3 \).

### Step-by-Step Solution:

1. **Given Information**:
We have:
\[ |a| = 5, \quad |b| = 4, \quad |c| = 3 \]
and
\[ a + b + c = 0 \]

2. **Dot Product of Zero Vector**:
Since \( a + b + c = 0 \), we can take the dot product of this equation with itself:
\[ (a + b + c) \cdot (a + b + c) = 0 \cdot 0 = 0 \]

3. **Expanding the Dot Product**:
Expanding the left-hand side, we get:
\[ a \cdot a + b \cdot b + c \cdot c + 2(a \cdot b + b \cdot c + c \cdot a) = 0 \]

4. **Substituting Magnitudes**:
We know that:
\[ a \cdot a = |a|^2 = 5^2 = 25, \quad b \cdot b = |b|^2 = 4^2 = 16, \quad c \cdot c = |c|^2 = 3^2 = 9 \]
Therefore, we can substitute these values into the equation:
\[ 25 + 16 + 9 + 2(a \cdot b + b \cdot c + c \cdot a) = 0 \]

5. **Combining Terms**:
Adding the constants:
\[ 25 + 16 + 9 = 50 \]
Thus, we have:
\[ 50 + 2(a \cdot b + b \cdot c + c \cdot a) = 0 \]

6. **Isolating the Dot Product**:
Rearranging the equation gives:
\[ 2(a \cdot b + b \cdot c + c \cdot a) = -50 \]

7. **Dividing by 2**:
Finally, we divide both sides by 2 to find:
\[ a \cdot b + b \cdot c + c \cdot a = -25 \]

### Final Answer:
The value of \( a \cdot b + b \cdot c + c \cdot a \) is \( \boxed{-25} \).