Given two points $M(3, -2)$ and $N(-5, -1)$, and the vector equation $\vec{MP} = \frac{1}{2}\vec{MN}$, find the coordinates of point $P$.

1. Problem Description

Given two points $M(3, -2)$ and $N(-5, -1)$, and the vector equation $\vec{MP} = \frac{1}{2}\vec{MN}$, find the coordinates of point $P$.

2. Solution Steps

Let the coordinates of point $P$ be $(x, y)$. The vector $\vec{MP}$ can be expressed as $(x - 3, y - (-2)) = (x - 3, y + 2)$.

The vector $\vec{MN}$ can be expressed as $(-5 - 3, -1 - (-2)) = (-8, 1)$.

Given that $\vec{MP} = \frac{1}{2}\vec{MN}$, we have

$(x - 3, y + 2) = \frac{1}{2}(-8, 1) = (-4, \frac{1}{2})$.

Equating the $x$-coordinates, we get $x - 3 = -4$, so $x = -4 + 3 = -1$.

Equating the $y$-coordinates, we get $y + 2 = \frac{1}{2}$, so $y = \frac{1}{2} - 2 = \frac{1}{2} - \frac{4}{2} = -\frac{3}{2}$.

Therefore, the coordinates of point $P$ are $(-1, -\frac{3}{2})$.

3. Final Answer

The coordinates of point $P$ are $(-1, -\frac{3}{2})$.