If `veca, vecb, vecc` are three vectors such that `veca + vecb+ vecc =0 and |veca| =5, |vecb|=12.|vecc|=13,` then find `vecavecb+vecb vec c + vec c veca`

Answer

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 \( |\vec{a}| = 5 \), \( |\vec{b}| = 12 \), and \( |\vec{c}| = 13 \).

### Step-by-Step Solution:

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

2. **Square both sides of the equation:**
\[ |\vec{a} + \vec{b} + \vec{c}|^2 = 0 \]
Expanding this using the dot product:
\[ (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{a} + \vec{b} + \vec{c}) = 0 \]
This expands to:
\[ \vec{a} \cdot \vec{a} + \vec{b} \cdot \vec{b} + \vec{c} \cdot \vec{c} + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0 \]

3. **Substitute the magnitudes:**
We know:
\[ |\vec{a}|^2 = 5^2 = 25, \quad |\vec{b}|^2 = 12^2 = 144, \quad |\vec{c}|^2 = 13^2 = 169 \]
Therefore, substituting these values:
\[ 25 + 144 + 169 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0 \]

4. **Calculate the sum of the squares:**
\[ 25 + 144 + 169 = 338 \]
So we have:
\[ 338 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0 \]

5. **Rearranging the equation:**
\[ 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = -338 \]
Dividing both sides by 2:
\[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = -169 \]

### Final Answer:
\[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = -169 \]