The problem asks to translate the given line segment by the vector $\begin{pmatrix} 4 \\ 2 \end{pmatrix}$. This means shifting the line segment 4 units to the right and 2 units up.

1. Problem Description

The problem asks to translate the given line segment by the vector $\begin{pmatrix} 4 \\ 2 \end{pmatrix}$. This means shifting the line segment 4 units to the right and 2 units up.

2. Solution Steps

The given line segment is a vertical line of length 4 units.

To translate this line segment by $\begin{pmatrix} 4 \\ 2 \end{pmatrix}$, we shift each point of the line segment 4 units to the right and 2 units up.

The bottom of the line segment is at $(x,y)$.

After the translation, the bottom of the line segment will be at $(x+4, y+2)$.

The top of the line segment is at $(x, y+4)$.

After the translation, the top of the line segment will be at $(x+4, y+4+2) = (x+4, y+6)$.

Therefore, the translated line segment is a vertical line segment of length 4, with bottom endpoint shifted 4 units to the right and 2 units up.

Since the original line segment lies at the bottom left corner, with its bottom point at $(1,1)$ and top point at $(1,5)$.

The bottom of the translated line segment is at $(1+4, 1+2)=(5,3)$.

The top of the translated line segment is at $(1+4, 5+2)=(5,7)$.

Therefore, the translated line segment is a vertical line segment with its bottom endpoint at $(5,3)$ and top endpoint at $(5,7)$.

3. Final Answer

The translated line segment is a vertical line segment of length 4 units, with its bottom at $(5,3)$. So the line spans from (5,3) to (5,7).

```

+---+---+---+---+---+---+

| | | | | | |

+---+---+---+---+---+---+

| | | | | x | |

+---+---+---+---+---+---+

| | | | | x | |

+---+---+---+---+---+---+

| | | | | x | |

+---+---+---+---+---+---+

| | | | | x | |

+---+---+---+---+---+---+

| | | | | | |

+---+---+---+---+---+---+

```