To solve for the vector $\vec{AB}$ in the given figure, we need to determine the coordinates of points $A$ and $B$, and then calculate the vector using these points.

Step-by-Step Solution:

1. Identify Coordinates of Points A and B:

   • Point $A$ is at $(1, -2)$.
   • Point $B$ is at $(4, -4)$.

2. Calculate the Vector $\vec{AB}$: The vector $\vec{AB}$ can be found by subtracting the coordinates of $A$ from $B$: $\vec{AB} = B - A = (4 - 1, -4 - (-2))$

3. Perform the Subtraction:

   • For the $x$-component: $4 - 1 = 3$
   • For the $y$-component: $-4 - (-2) = -4 + 2 = -2$

   Therefore: $\vec{AB} = (3, -2)$

Final Answer:

The vector $\vec{AB}$ is $(3, -2)$.

Would you like further details on how vectors are calculated or have any questions on this topic?

Here are some related questions you might find useful:

1. What is the magnitude of vector $\vec{AB}$?
2. How would you find the direction angle of $\vec{AB}$?
3. What are unit vectors, and how can we find the unit vector in the direction of $\vec{AB}$?
4. How would the vector calculation change if $A$ and $B$ were given in three dimensions?
5. How can vectors be used to determine the displacement between two points?

Tip: When calculating a vector between two points, always subtract the initial point coordinates from the terminal point coordinates.