Three vectors ` vec( a)`, ` vec(b)` and ` vec(c)` satisfy the condition ` vec(a)+ vec( b)+ vec(c)= vec(0)`. Evaluate the quantity `mu= vec(a).vec( b)+ vec(b).vec( c)+ vec(c).vec(a)`, if `| vec(a)|=1,| vec(b)|=4`and `| vec(c)|=2`.

Explanation

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 of the vectors \( |\vec{a}| = 1 \), \( |\vec{b}| = 4 \), and \( |\vec{c}| = 2 \).

### Step-by-Step Solution:

1. **Use the condition \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \)**:
From this condition, we can express one vector in terms of the others. Let's express \( \vec{c} \):
\[
\vec{c} = -(\vec{a} + \vec{b})
...