If `|veca| =5, |vecb| =4, |vecc| =3 and veca + vecb + vecc =0,` then the vlaue of `|veca, vecb + vecb.vecc+vecc,veca|,` is

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 that \(|\vec{a}| = 5\), \(|\vec{b}| = 4\), \(|\vec{c}| = 3\), and \(\vec{a} + \vec{b} + \vec{c} = 0\).

### Step 1: Understand the given information
We have three vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) such that their sum is zero. This implies that \(\vec{c} = -(\vec{a} + \vec{b})\).

### Step 2: Use the dot product
Taking the dot product of \(\vec{a} + \vec{b} + \vec{c} = 0\) with \(\vec{a}\):
\[ \vec{a} \cdot (\vec{a} + \vec{b} + \vec{c}) = 0 \]
This expands to:
\[ \vec{a} \cdot \vec{a} + \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = 0 \]
Substituting \(|\vec{a}|^2\) for \(\vec{a} \cdot \vec{a}\):
\[ |\vec{a}|^2 + \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = 0 \]
This gives us:
\[ 25 + \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = 0 \]
Thus,
\[ \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = -25 \quad \text{(Equation 1)} \]

### Step 3: Take the dot product with \(\vec{b}\)
Now, taking the dot product with \(\vec{b}\):
\[ \vec{b} \cdot (\vec{a} + \vec{b} + \vec{c}) = 0 \]
Expanding this gives:
\[ \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{b} + \vec{b} \cdot \vec{c} = 0 \]
Substituting \(|\vec{b}|^2\):
\[ \vec{b} \cdot \vec{a} + 16 + \vec{b} \cdot \vec{c} = 0 \]
Thus,
\[ \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{c} = -16 \quad \text{(Equation 2)} \]

### Step 4: Take the dot product with \(\vec{c}\)
Now, taking the dot product with \(\vec{c}\):
\[ \vec{c} \cdot (\vec{a} + \vec{b} + \vec{c}) = 0 \]
Expanding this gives:
\[ \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} + \vec{c} \cdot \vec{c} = 0 \]
Substituting \(|\vec{c}|^2\):
\[ \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} + 9 = 0 \]
Thus,
\[ \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} = -9 \quad \text{(Equation 3)} \]

### Step 5: Combine the equations
Now we have three equations:
1. \(\vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = -25\)
2. \(\vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{c} = -16\)
3. \(\vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} = -9\)

Adding these three equations:
\[ (\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) + (\vec{a} \cdot \vec{c} + \vec{b} \cdot \vec{a} + \vec{c} \cdot \vec{b}) = -25 - 16 - 9 \]
This simplifies to:
\[ 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = -50 \]
Thus,
\[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = -25 \]

### Step 6: Find the magnitude
The magnitude we need is:
\[ |\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}| = |-25| = 25 \]

### Final Answer
The value of \(|\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}|\) is **25**.