If the three vectors a, b and c with magnitude 3, 4 and 5 respectively and `a+b+c= 0`, then the value of a.b+b. c+c a is

Options

A. `-23`
B. `-25`
C. 30
D. 26

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

### Step-by-Step Solution:

1. **Understanding the Given Information:**
We know that \( a + b + c = 0 \). This implies that \( c = - (a + b) \).

2. **Using the Magnitudes:**
The magnitudes of the vectors are given as:
\[ |a| = 3, \quad |b| = 4, \quad |c| = 5 \]
Therefore:
\[ |a|^2 = 9, \quad |b|^2 = 16, \quad |c|^2 = 25 \]

3. **Finding \( |c|^2 \) in terms of \( a \) and \( b \):**
Since \( c = - (a + b) \), we can find \( |c|^2 \):
\[ |c|^2 = |-(a + b)|^2 = |a + b|^2 \]
Expanding \( |a + b|^2 \):
\[ |a + b|^2 = |a|^2 + |b|^2 + 2(a \cdot b) \]
Substituting the magnitudes:
\[ 25 = 9 + 16 + 2(a \cdot b) \]
Simplifying this gives:
\[ 25 = 25 + 2(a \cdot b) \implies 2(a \cdot b) = 0 \implies a \cdot b = 0 \]

4. **Finding \( b \cdot c \) and \( c \cdot a \):**
Now, we can express \( b \cdot c \) and \( c \cdot a \) using \( c = - (a + b) \):
\[ b \cdot c = b \cdot (- (a + b)) = - (b \cdot a + b \cdot b) = - (0 + 16) = -16 \]
\[ c \cdot a = c \cdot (- (a + b)) = - (c \cdot a + c \cdot b) = - (c \cdot a + (-16)) = - (c \cdot a - 16) \]
To find \( c \cdot a \), we can use the fact that \( c = - (a + b) \):
\[ c \cdot a = - (a + b) \cdot a = - (a \cdot a + b \cdot a) = - (9 + 0) = -9 \]

5. **Calculating \( a \cdot b + b \cdot c + c \cdot a \):**
Now we can substitute these values into the expression:
\[ a \cdot b + b \cdot c + c \cdot a = 0 - 16 - 9 \]
Simplifying this gives:
\[ a \cdot b + b \cdot c + c \cdot a = -25 \]

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