Given the vectors $\vec{OM} = (3, 1)$ and $\vec{ON} = (-5, -1)$, find the vector $\frac{1}{2}\vec{MN}$.

1. Problem Description

Given the vectors $\vec{OM} = (3, 1)$ and $\vec{ON} = (-5, -1)$, find the vector $\frac{1}{2}\vec{MN}$.

2. Solution Steps

First, we need to find the vector $\vec{MN}$. The vector $\vec{MN}$ can be expressed as $\vec{ON} - \vec{OM}$.

$$\vec{MN} = \vec{ON} - \vec{OM}$$

Substituting the given vectors, we have

$$\vec{MN} = (-5, -1) - (3, 1)$$

$$\vec{MN} = (-5 - 3, -1 - 1)$$

$$\vec{MN} = (-8, -2)$$

Now, we need to find $\frac{1}{2}\vec{MN}$.

$$\frac{1}{2}\vec{MN} = \frac{1}{2}(-8, -2)$$

$$\frac{1}{2}\vec{MN} = (\frac{1}{2} \times -8, \frac{1}{2} \times -2)$$

$$\frac{1}{2}\vec{MN} = (-4, -1)$$

3. Final Answer

$\frac{1}{2}\vec{MN} = (-4, -1)$