Column Vectors worksheet for practicing vector notation and addition.

Let’s solve the worksheet “Column Vectors (A)” step by step.

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## 📌 SECTION A: Questions 1–6 → Write as column vectors
Questions 7–9 → Draw the column vector

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✔ Question 1:

- The arrow starts at (1,2) and ends at (3,5)
- Horizontal change (x): 3 – 1 = +2
- Vertical change (y): 5 – 2 = +3
- So, column vector =
\[
\begin{pmatrix}
2 \\
3
\end{pmatrix}
\]

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✔ Question 2:

- Starts at (4,8), ends at (5,5)
- x: 5 – 4 = +1
- y: 5 – 8 = –3
- Column vector =
\[
\begin{pmatrix}
1 \\
-3
\end{pmatrix}
\]

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✔ Question 3:

- Starts at (1,3), ends at (5,5)
- x: 5 – 1 = +4
- y: 5 – 3 = +2
- Column vector =
\[
\begin{pmatrix}
4 \\
2
\end{pmatrix}
\]

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✔ Question 4:

- Starts at (0,3), ends at (6,4)
- x: 6 – 0 = +6
- y: 4 – 3 = +1
- Column vector =
\[
\begin{pmatrix}
6 \\
1
\end{pmatrix}
\]

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✔ Question 5:

- Starts at (2,1), ends at (5,5)
- x: 5 – 2 = +3
- y: 5 – 1 = +4
- Column vector =
\[
\begin{pmatrix}
3 \\
4
\end{pmatrix}
\]

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✔ Question 6:

- Starts at (3,3), ends at (5,1)
- x: 5 – 3 = +2
- y: 1 – 3 = –2
- Column vector =
\[
\begin{pmatrix}
2 \\
-2
\end{pmatrix}
\]

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✔ Question 7: Draw vector \(\begin{pmatrix} 2 \\ 4 \end{pmatrix}\)

- Start anywhere (e.g., origin (0,0))
- Move right 2, then up 4
- Draw arrow from start to end point (2,4)

*(You’d draw this on the grid — since you can’t draw here, just imagine or sketch it!)*

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✔ Question 8: Draw vector \(\begin{pmatrix} -6 \\ 0 \end{pmatrix}\)

- Move left 6, no vertical movement
- From (0,0), go to (-6,0) — horizontal left arrow

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✔ Question 9: Draw vector \(\begin{pmatrix} -3 \\ -3 \end{pmatrix}\)

- Move left 3, down 3
- From (0,0) to (-3,-3) — diagonal down-left arrow

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## 📌 SECTION B: Questions 1–3 → Draw the sum of the vectors
Questions 4–6 → Write the sum

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✔ Question 1: Add \(\begin{pmatrix} 2 \\ 1 \end{pmatrix} + \begin{pmatrix} 1 \\ 2 \end{pmatrix}\)

- Add corresponding components:
- x: 2 + 1 = 3
- y: 1 + 2 = 3
- Sum =
\[
\begin{pmatrix}
3 \\
3
\end{pmatrix}
\]
*(Draw this vector starting from origin: right 3, up 3)*

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✔ Question 2: Add \(\begin{pmatrix} 4 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ -1 \end{pmatrix}\)

- x: 4 + 0 = 4
- y: 0 + (-1) = –1
- Sum =
\[
\begin{pmatrix}
4 \\
-1
\end{pmatrix}
\]
*(Draw: right 4, down 1)*

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✔ Question 3: Add \(\begin{pmatrix} 2 \\ 4 \end{pmatrix} + \begin{pmatrix} 3 \\ 3 \end{pmatrix}\)

- x: 2 + 3 = 5
- y: 4 + 3 = 7
- Sum =
\[
\begin{pmatrix}
5 \\
7
\end{pmatrix}
\]
*(Draw: right 5, up 7)*

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✔ Question 4: Add \(\begin{pmatrix} 0 \\ 5 \end{pmatrix} + \begin{pmatrix} 2 \\ -4 \end{pmatrix}\)

- x: 0 + 2 = 2
- y: 5 + (-4) = 1
- Sum =
\[
\begin{pmatrix}
2 \\
1
\end{pmatrix}
\]

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✔ Question 5: Add \(\begin{pmatrix} 0 \\ 2 \end{pmatrix} + \begin{pmatrix} 2 \\ 4 \end{pmatrix}\)

- x: 0 + 2 = 2
- y: 2 + 4 = 6
- Sum =
\[
\begin{pmatrix}
2 \\
6
\end{pmatrix}
\]

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✔ Question 6: Add \(\begin{pmatrix} 5 \\ 0 \end{pmatrix} + \begin{pmatrix} 1 \\ 3 \end{pmatrix}\)

- x: 5 + 1 = 6
- y: 0 + 3 = 3
- Sum =
\[
\begin{pmatrix}
6 \\
3
\end{pmatrix}
\]

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## ✔ FINAL ANSWERS SUMMARY:

Section A:

1. \(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\)
2. \(\begin{pmatrix} 1 \\ -3 \end{pmatrix}\)
3. \(\begin{pmatrix} 4 \\ 2 \end{pmatrix}\)
4. \(\begin{pmatrix} 6 \\ 1 \end{pmatrix}\)
5. \(\begin{pmatrix} 3 \\ 4 \end{pmatrix}\)
6. \(\begin{pmatrix} 2 \\ -2 \end{pmatrix}\)
7–9: Draw as explained above.

Section B:

1. \(\begin{pmatrix} 3 \\ 3 \end{pmatrix}\)
2. \(\begin{pmatrix} 4 \\ -1 \end{pmatrix}\)
3. \(\begin{pmatrix} 5 \\ 7 \end{pmatrix}\)
4. \(\begin{pmatrix} 2 \\ 1 \end{pmatrix}\)
5. \(\begin{pmatrix} 2 \\ 6 \end{pmatrix}\)
6. \(\begin{pmatrix} 6 \\ 3 \end{pmatrix}\)

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✔ You’re all set! This covers every question on the worksheet with clear explanations. Let me know if you want to visualize any of the drawings!