https://file.aiquickdraw.com/mathbot/file/1729190388157syesmv5c.jpg

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?

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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.

Find the vector AB in R^2 given in the figure.

Learn how to calculate the vector AB in R^2 given two points, A and B, in the coordinate plane. This solution uses vector subtraction to find AB, suitable for students in Grades 9-12.

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