Geometrical Proof with Vectors - Mr-Mathematics.com - Free Printable

To solve the problem of describing translations using column vectors, we need to determine how each shape has been moved horizontally and vertically from its original position (Shape A) to its new position. The column vector will be in the form:

\[
\begin{pmatrix}
a \\
b
\end{pmatrix}
\]

where \( a \) represents the horizontal movement (right if positive, left if negative) and \( b \) represents the vertical movement (up if positive, down if negative).

Let's analyze each question step by step:

---

1)

- Original Shape A is at the left.
- Translated Shape is to the right.
- Horizontal movement: +4 (4 units to the right).
- Vertical movement: 0 (no vertical change).
- Column vector:
\[
\begin{pmatrix}
4 \\
0
\end{pmatrix}
\]

---

2)

- Original Shape A is at the top.
- Translated Shape is below.
- Horizontal movement: 0 (no horizontal change).
- Vertical movement: -3 (3 units down).
- Column vector:
\[
\begin{pmatrix}
0 \\
-3
\end{pmatrix}
\]

---

3)

- Original Shape A is at the top-left.
- Translated Shape is to the right and below.
- Horizontal movement: +3 (3 units to the right).
- Vertical movement: -2 (2 units down).
- Column vector:
\[
\begin{pmatrix}
3 \\
-2
\end{pmatrix}
\]

---

4)

- Original Shape A is at the bottom-left.
- Translated Shape is to the right.
- Horizontal movement: +5 (5 units to the right).
- Vertical movement: 0 (no vertical change).
- Column vector:
\[
\begin{pmatrix}
5 \\
0
\end{pmatrix}
\]

---

5)

- Original Shape A is at the right.
- Translated Shape is to the left.
- Horizontal movement: -3 (3 units to the left).
- Vertical movement: 0 (no vertical change).
- Column vector:
\[
\begin{pmatrix}
-3 \\
0
\end{pmatrix}
\]

---

6)

- Original Shape A is at the top-left.
- Translated Shape is to the right and slightly down.
- Horizontal movement: +4 (4 units to the right).
- Vertical movement: -1 (1 unit down).
- Column vector:
\[
\begin{pmatrix}
4 \\
-1
\end{pmatrix}
\]

---

7)

- Original Shape A is at the bottom-right.
- Translated Shape is to the left and up.
- Horizontal movement: -3 (3 units to the left).
- Vertical movement: +4 (4 units up).
- Column vector:
\[
\begin{pmatrix}
-3 \\
4
\end{pmatrix}
\]

---

8)

- Original Shape A is at the center.
- Translated Shape is to the right and slightly up.
- Horizontal movement: +3 (3 units to the right).
- Vertical movement: +2 (2 units up).
- Column vector:
\[
\begin{pmatrix}
3 \\
2
\end{pmatrix}
\]

---

9)

- Original Shape A is at the top-left.
- Translated Shape is to the right and down.
- Horizontal movement: +5 (5 units to the right).
- Vertical movement: -3 (3 units down).
- Column vector:
\[
\begin{pmatrix}
5 \\
-3
\end{pmatrix}
\]

---

Final Answers:

\[
\boxed{
\begin{array}{ccc}
1) & \begin{pmatrix} 4 \\ 0 \end{pmatrix} & 2) & \begin{pmatrix} 0 \\ -3 \end{pmatrix} & 3) & \begin{pmatrix} 3 \\ -2 \end{pmatrix} \\
4) & \begin{pmatrix} 5 \\ 0 \end{pmatrix} & 5) & \begin{pmatrix} -3 \\ 0 \end{pmatrix} & 6) & \begin{pmatrix} 4 \\ -1 \end{pmatrix} \\
7) & \begin{pmatrix} -3 \\ 4 \end{pmatrix} & 8) & \begin{pmatrix} 3 \\ 2 \end{pmatrix} & 9) & \begin{pmatrix} 5 \\ -3 \end{pmatrix}
\end{array}
}
\]