If `vec(a)+vec(b)+vec(c )=0 " and" |vec(a)|=3,|vec(b)|=5, |vec(c )|=7`, then find the value of `vec(a),vec(b)+vec(b).vec(c )+vec(c ).vec(a)`.

Explanation

To solve the problem, we start with the given conditions:

1. \(\vec{a} + \vec{b} + \vec{c} = 0\)
2. \(|\vec{a}| = 3\)
3. \(|\vec{b}| = 5\)
4. \(|\vec{c}| = 7\)

We need to find the value of \(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}\).

### Step 1: Rewrite \(\vec{c}\)
From the first equation, we can express \(\vec{c}\) in terms of \(\vec{a}\) and \(\vec{b}\):
\[ \vec{c} = -(\vec{a} + \vec{b}) \]

### Step 2: Calculate \(|\vec{c}|\)
Using the magnitude of \(\vec{c}\):
\[ |\vec{c}| = |-(\vec{a} + \vec{b})| = |\vec{a} + \vec{b}| \]
Since \(|\vec{c}| = 7\), we have:
\[ |\vec{a} + \vec{b}| = 7 \]

### Step 3: Use the formula for the magnitude of a vector sum
Using the formula for the magnitude of the sum of two vectors:
\[ |\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2 \vec{a} \cdot \vec{b} \]
Substituting the known magnitudes:
\[ 7^2 = 3^2 + 5^2 + 2 \vec{a} \cdot \vec{b} \]
Calculating the squares:
\[ 49 = 9 + 25 + 2 \vec{a} \cdot \vec{b} \]
\[ 49 = 34 + 2 \vec{a} \cdot \vec{b} \]
\[ 2 \vec{a} \cdot \vec{b} = 49 - 34 = 15 \]
\[ \vec{a} \cdot \vec{b} = \frac{15}{2} \]

### Step 4: Calculate \(\vec{b} \cdot \vec{c}\)
Now, we can calculate \(\vec{b} \cdot \vec{c}\):
\[ \vec{c} = -(\vec{a} + \vec{b}) \implies \vec{b} \cdot \vec{c} = \vec{b} \cdot -(\vec{a} + \vec{b}) = -(\vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{b}) \]
Substituting the values we have:
\[ \vec{b} \cdot \vec{c} = -\left(\frac{15}{2} + 25\right) = -\left(\frac{15}{2} + \frac{50}{2}\right) = -\frac{65}{2} \]

### Step 5: Calculate \(\vec{c} \cdot \vec{a}\)
Next, we calculate \(\vec{c} \cdot \vec{a}\):
\[ \vec{c} \cdot \vec{a} = -(\vec{a} + \vec{b}) \cdot \vec{a} = -(\vec{a} \cdot \vec{a} + \vec{b} \cdot \vec{a}) \]
Substituting the values:
\[ \vec{c} \cdot \vec{a} = -\left(9 + \frac{15}{2}\right) = -\left(9 + 7.5\right) = -16.5 = -\frac{33}{2} \]

### Step 6: Combine the results
Now we can combine all the dot products:
\[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = \frac{15}{2} - \frac{65}{2} - \frac{33}{2} \]
Calculating:
\[ = \frac{15 - 65 - 33}{2} = \frac{-83}{2} \]

### Final Answer
Thus, the value of \(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}\) is:
\[ \boxed{-\frac{83}{2}} \]