If the vectors a,b,c satisfy the condition `a+b+c =0,` the value of `a.b +b.c +c.a` if `|a| =1,|b| =3` and `|c| =4` is

Options

A. 11
B. `-13`
C. 15
D. 12

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 \( |a| = 1 \), \( |b| = 3 \), and \( |c| = 4 \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ a + b + c = 0 \]
This implies:
\[ c = - (a + b) \]

2. **Take the dot product of both sides with vector \( a \)**:
\[ a \cdot (a + b + c) = a \cdot 0 \]
This simplifies to:
\[ a \cdot a + a \cdot b + a \cdot c = 0 \]
Since \( a \cdot a = |a|^2 = 1^2 = 1 \), we have:
\[ 1 + a \cdot b + a \cdot c = 0 \]
Rearranging gives:
\[ a \cdot b + a \cdot c = -1 \quad \text{(Equation 1)} \]

3. **Now take the dot product of both sides with vector \( b \)**:
\[ b \cdot (a + b + c) = b \cdot 0 \]
This simplifies to:
\[ b \cdot a + b \cdot b + b \cdot c = 0 \]
Since \( b \cdot b = |b|^2 = 3^2 = 9 \), we have:
\[ a \cdot b + 9 + b \cdot c = 0 \]
Rearranging gives:
\[ a \cdot b + b \cdot c = -9 \quad \text{(Equation 2)} \]

4. **Next, take the dot product of both sides with vector \( c \)**:
\[ c \cdot (a + b + c) = c \cdot 0 \]
This simplifies to:
\[ c \cdot a + c \cdot b + c \cdot c = 0 \]
Since \( c \cdot c = |c|^2 = 4^2 = 16 \), we have:
\[ c \cdot a + c \cdot b + 16 = 0 \]
Rearranging gives:
\[ c \cdot a + b \cdot c = -16 \quad \text{(Equation 3)} \]

5. **Now we have three equations**:
- From Equation 1: \( a \cdot b + a \cdot c = -1 \)
- From Equation 2: \( a \cdot b + b \cdot c = -9 \)
- From Equation 3: \( c \cdot a + b \cdot c = -16 \)

6. **Add all three equations together**:
\[ (a \cdot b + a \cdot c) + (a \cdot b + b \cdot c) + (c \cdot a + b \cdot c) = -1 - 9 - 16 \]
This simplifies to:
\[ 2(a \cdot b) + 2(b \cdot c) + 2(c \cdot a) = -26 \]
Dividing by 2 gives:
\[ a \cdot b + b \cdot c + c \cdot a = -13 \]

### Final Answer:
\[ a \cdot b + b \cdot c + c \cdot a = -13 \]