The problem asks to translate the given line segment by the given vectors in parts a and b. Part a: Translate by the vector $\begin{pmatrix} 1 \\ -2 \end{pmatrix}$. Part b: Translate by the vector $\begin{pmatrix} -1 \\ -2 \end{pmatrix}$.

1. Problem Description

The problem asks to translate the given line segment by the given vectors in parts a and b.

Part a: Translate by the vector $\begin{pmatrix} 1 \\ -2 \end{pmatrix}$.

Part b: Translate by the vector $\begin{pmatrix} -1 \\ -2 \end{pmatrix}$.

2. Solution Steps

Part a:

The vector $\begin{pmatrix} 1 \\ -2 \end{pmatrix}$ means we move the line segment 1 unit to the right and 2 units down.

Original line segment coordinates are approximately (1, 1) to (1, 4).

New coordinates will be (1+1, 1-2) to (1+1, 4-2), which is (2, -1) to (2, 2).

Part b:

The vector $\begin{pmatrix} -1 \\ -2 \end{pmatrix}$ means we move the line segment 1 unit to the left and 2 units down.

Original line segment coordinates are approximately (1, 1) to (1, 4).

New coordinates will be (1-1, 1-2) to (1-1, 4-2), which is (0, -1) to (0, 2).

3. Final Answer

Part a: The line segment after translation will have coordinates (2, -1) to (2, 2). Visually, it would be the vertical line going from y = -1 to y = 2 at x =

2.

Part b: The line segment after translation will have coordinates (0, -1) to (0, 2). Visually, it would be the vertical line going from y = -1 to y = 2 at x =

0.