` vec a , vec b , vec c ` are three vectors such that ` |vec a | = 5 | vec b | =8 |vec c | = 3 if vec a + vec b + vec c = 8 hati+ 6hatj , `find the value of ` vec a .vec b +vec b .vec c + vec c .vec a `

Answer

To solve the problem step by step, we will follow the given information and apply vector algebra.

### Step 1: Understand the given information
We have three vectors:
- \( |\vec{a}| = 5 \)
- \( |\vec{b}| = 8 \)
- \( |\vec{c}| = 3 \)

And we know that:
\[ \vec{a} + \vec{b} + \vec{c} = 8\hat{i} + 6\hat{j} \]

### Step 2: Square both sides of the equation
To find \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \), we will square both sides of the equation:
\[ |\vec{a} + \vec{b} + \vec{c}|^2 = |8\hat{i} + 6\hat{j}|^2 \]

### Step 3: Calculate the left-hand side
Using the property of magnitudes:
\[ |\vec{a} + \vec{b} + \vec{c}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) \]
Substituting the magnitudes:
\[ |\vec{a}|^2 = 5^2 = 25, \quad |\vec{b}|^2 = 8^2 = 64, \quad |\vec{c}|^2 = 3^2 = 9 \]
Thus,
\[ |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 = 25 + 64 + 9 = 98 \]

### Step 4: Calculate the right-hand side
Now, calculate the magnitude of the right-hand side:
\[ |8\hat{i} + 6\hat{j}|^2 = 8^2 + 6^2 = 64 + 36 = 100 \]

### Step 5: Set the equations equal
Now we equate both sides:
\[ 98 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 100 \]

### Step 6: Solve for the dot products
Rearranging gives:
\[ 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 100 - 98 \]
\[ 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 2 \]
Dividing by 2:
\[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = 1 \]

### Final Answer
Thus, the value of \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \) is:
\[ \boxed{1} \]