Three vectors ` vec a ,\ vec b ,\ vec c` satisfy the condition ` vec a+ vec b+ vec c= vec0`. Evaluate the quantity `mu= vec (a)cdot vec (b)+ vec (b )cdot vec (c)+ vec (c)cdot vec (a) ,\ if\ | vec a|=1,\ | vec b|=4\ a n d\ | vec c|=2.`

Options

A. `-27/2`
B. `-21/2`
C. `-29/2`
D. `-19/2`

Answer

To solve the problem, we need to evaluate the quantity \( \mu = \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \) given the condition \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \) and the magnitudes \( |\vec{a}| = 1 \), \( |\vec{b}| = 4 \), and \( |\vec{c}| = 2 \).

### Step-by-step Solution:

1. **Write the given condition**:
\[ \vec{a} + \vec{b} + \vec{c} = \vec{0} \]

2. **Dot the equation with \( \vec{a} \)**:
\[ \vec{a} \cdot (\vec{a} + \vec{b} + \vec{c}) = \vec{a} \cdot \vec{0} \]
This simplifies to:
\[ \vec{a} \cdot \vec{a} + \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = 0 \]
Since \( |\vec{a}| = 1 \), we have:
\[ 1 + \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = 0 \]
Thus, we can express:
\[ \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = -1 \quad \text{(Equation 1)} \]

3. **Dot the equation with \( \vec{b} \)**:
\[ \vec{b} \cdot (\vec{a} + \vec{b} + \vec{c}) = \vec{b} \cdot \vec{0} \]
This simplifies to:
\[ \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{b} + \vec{b} \cdot \vec{c} = 0 \]
Since \( |\vec{b}| = 4 \), we have:
\[ \vec{b} \cdot \vec{a} + 16 + \vec{b} \cdot \vec{c} = 0 \]
Thus, we can express:
\[ \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{c} = -16 \quad \text{(Equation 2)} \]

4. **Dot the equation with \( \vec{c} \)**:
\[ \vec{c} \cdot (\vec{a} + \vec{b} + \vec{c}) = \vec{c} \cdot \vec{0} \]
This simplifies to:
\[ \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} + \vec{c} \cdot \vec{c} = 0 \]
Since \( |\vec{c}| = 2 \), we have:
\[ \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} + 4 = 0 \]
Thus, we can express:
\[ \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} = -4 \quad \text{(Equation 3)} \]

5. **Add the three equations**:
Adding Equation 1, Equation 2, and Equation 3:
\[ (\vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}) + (\vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{c}) + (\vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b}) = -1 - 16 - 4 \]
This simplifies to:
\[ 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = -21 \]

6. **Solve for \( \mu \)**:
\[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = \frac{-21}{2} \]
Therefore, the value of \( \mu \) is:
\[ \mu = -\frac{21}{2} \]

### Final Answer:
\[ \mu = -\frac{21}{2} \]