If `veca,vecbandvecc` are three vectors such that `veca+vecb+vecc=0and|veca|=2,|vecb|=3and|vecc|=5`, then the value of `veca.vecb+vecb.vecc+vecc.veca` is

Options

A. 0
B. 1
C. `-19`
D. 38

Explanation

To solve the problem, we need to find the value of \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \) given the conditions \( \vec{a} + \vec{b} + \vec{c} = 0 \) and the magnitudes \( |\vec{a}| = 2 \), \( |\vec{b}| = 3 \), and \( |\vec{c}| = 5 \).

### Step-by-Step Solution

1. **Understanding the Given Information:**
We have three vectors \( \vec{a}, \vec{b}, \vec{c} \) such that:
\[ \vec{a} + \vec{b} + \vec{c} = 0 \]
This implies:
\[ \vec{c} = -(\vec{a} + \vec{b}) \]

2. **Squaring the Magnitudes:**
We know the magnitudes of the vectors:
\[ |\vec{a}|^2 = 4, \quad |\vec{b}|^2 = 9, \quad |\vec{c}|^2 = 25 \]

3. **Taking the Dot Product:**
Taking the dot product of the equation \( \vec{a} + \vec{b} + \vec{c} = 0 \) with itself:
\[ (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{a} + \vec{b} + \vec{c}) = 0 \]
Expanding this gives:
\[ \vec{a} \cdot \vec{a} + \vec{b} \cdot \vec{b} + \vec{c} \cdot \vec{c} + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0 \]

4. **Substituting the Magnitudes:**
Substitute the known magnitudes:
\[ |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 = 4 + 9 + 25 = 38 \]
Therefore, we have:
\[ 38 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0 \]

5. **Isolating the Dot Product Terms:**
Rearranging gives:
\[ 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = -38 \]
Dividing both sides by 2:
\[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = -19 \]

### Final Answer
The value of \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \) is:
\[ \boxed{-19} \]