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. **Use the given magnitudes to find the squares:**
\[ |\vec{a}|^2 = 2^2 = 4, \quad |\vec{b}|^2 = 3^2 = 9, \quad |\vec{c}|^2 = 5^2 = 25 \]

2. **Substitute into the equation:**
We know that \( \vec{a} + \vec{b} + \vec{c} = 0 \). Taking the dot product of both sides with itself gives:
\[ (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{a} + \vec{b} + \vec{c}) = 0 \]

3. **Expand the dot product:**
\[ \vec{a} \cdot \vec{a} + \vec{b} \cdot \vec{b} + \vec{c} \cdot \vec{c} + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0 \]

4. **Substitute the values:**
\[ 4 + 9 + 25 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0 \]

5. **Combine the constants:**
\[ 38 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0 \]

6. **Isolate the dot product term:**
\[ 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = -38 \]

7. **Divide by 2:**
\[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = -19 \]

### Final Answer:
\[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = -19 \]