Comprehensive worksheet for practicing column vector addition through diagrams and calculations.

Let's solve the Column Vector Addition worksheet step by step, explaining both Section A and Section B.

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🔷 Section A: Interpret Diagrams to Find Resultant Vectors

We are given diagrams showing two vectors (one blue, one green) added together using the head-to-tail method, and we must find the resultant vector (the red vector), written as a column vector.

> 📌 Rule: To add two vectors graphically:
> - Place the tail of the second vector at the head of the first.
> - The resultant is from the tail of the first to the head of the second.
> - Read horizontal (x) and vertical (y) components from the grid.

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#### ✔ Example 1 (Given):
- Blue vector: moves 3 right → x = +3
- Green vector: moves 2 up → y = +2
- Resultant:
$$
\begin{pmatrix} 3 \\ 2 \end{pmatrix}
$$

Now let’s do each question:

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#### 2.
- Blue vector: moves 2 right → x = +2
- Green vector: moves 4 up → y = +4
- Resultant:
$$
\begin{pmatrix} 2 \\ 4 \end{pmatrix}
$$

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#### 3.
- Blue vector: moves 0 right, 3 up → x = 0, y = +3
- Green vector: moves 4 right, 0 up → x = +4, y = 0
- Resultant:
$$
\begin{pmatrix} 4 \\ 3 \end{pmatrix}
$$

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#### 4.
- Blue vector: moves 2 left, 3 up → x = –2, y = +3
- Green vector: moves 5 right, 0 up → x = +5, y = 0
- Resultant:
$$
(-2 + 5, 3 + 0) = \begin{pmatrix} 3 \\ 3 \end{pmatrix}
$$

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#### 5.
- Blue vector: moves 3 right, 3 up → x = +3, y = +3
- Green vector: moves 3 right, 3 down → x = +3, y = –3
- Resultant:
$$
(3+3, 3–3) = \begin{pmatrix} 6 \\ 0 \end{pmatrix}
$$

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#### 6.
- Blue vector: moves 0 right, 4 up → x = 0, y = +4
- Green vector: moves 3 right, 2 up → x = +3, y = +2
- Resultant:
$$
\begin{pmatrix} 3 \\ 6 \end{pmatrix}
$$

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#### 7.
- Blue vector: moves 3 right, 0 up → x = +3, y = 0
- Green vector: moves 0 right, 4 down → x = 0, y = –4
- Resultant:
$$
\begin{pmatrix} 3 \\ -4 \end{pmatrix}
$$

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#### 8.
- Blue vector: moves 4 left, 0 up → x = –4, y = 0
- Green vector: moves 0 right, 3 down → x = 0, y = –3
- Resultant:
$$
\begin{pmatrix} -4 \\ -3 \end{pmatrix}
$$

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#### 9.
- Blue vector: moves 3 right, 3 up → x = +3, y = +3
- Green vector: moves 3 right, 3 down → x = +3, y = –3
- Resultant:
$$
(3+3, 3–3) = \begin{pmatrix} 6 \\ 0 \end{pmatrix}
$$

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✔ Section A Answers Summary:

| Q | Resultant Vector |
|---|------------------|
| 1 | $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$ (given) |
| 2 | $\begin{pmatrix} 2 \\ 4 \end{pmatrix}$ |
| 3 | $\begin{pmatrix} 4 \\ 3 \end{pmatrix}$ |
| 4 | $\begin{pmatrix} 3 \\ 3 \end{pmatrix}$ |
| 5 | $\begin{pmatrix} 6 \\ 0 \end{pmatrix}$ |
| 6 | $\begin{pmatrix} 3 \\ 6 \end{pmatrix}$ |
| 7 | $\begin{pmatrix} 3 \\ -4 \end{pmatrix}$ |
| 8 | $\begin{pmatrix} -4 \\ -3 \end{pmatrix}$ |
| 9 | $\begin{pmatrix} 6 \\ 0 \end{pmatrix}$ |

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🔷 Section B: Draw Diagrams for Given Vector Additions

We are given two column vectors to add. We need to:
1. Add them algebraically.
2. Draw a diagram showing the addition using the head-to-tail method.

> 📌 Rule:
> $$
> \begin{pmatrix} a \\ b \end{pmatrix} + \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a+c \\ b+d \end{pmatrix}
> $$

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#### Example (Given):
$$
\begin{pmatrix} 3 \\ 1 \end{pmatrix} + \begin{pmatrix} 0 \\ 2 \end{pmatrix} = \begin{pmatrix} 3 \\ 3 \end{pmatrix}
$$
✔ Already shown in diagram.

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#### 2.
$$
\begin{pmatrix} 1 \\ 3 \end{pmatrix} + \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}
$$

📌 How to draw:
- Start at origin.
- Draw first vector: move 1 right, 3 up.
- From its tip, draw second vector: 1 right, 1 up.
- Resultant: from origin to final point: (2, 4)

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#### 3.
$$
\begin{pmatrix} 0 \\ 2 \end{pmatrix} + \begin{pmatrix} 2 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}
$$

📌 Draw:
- First vector: straight up 2 units.
- Then: 2 right, 2 up.
- Resultant: from start to end → (2, 4)

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#### 4.
$$
\begin{pmatrix} 4 \\ 1 \end{pmatrix} + \begin{pmatrix} -3 \\ 2 \end{pmatrix} = \begin{pmatrix} 1 \\ 3 \end{pmatrix}
$$

📌 Draw:
- First: 4 right, 1 up.
- Second: 3 left, 2 up.
- Resultant: 1 right, 3 up → (1, 3)

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#### 5.
$$
\begin{pmatrix} -2 \\ -2 \end{pmatrix} + \begin{pmatrix} 3 \\ -1 \end{pmatrix} = \begin{pmatrix} 1 \\ -3 \end{pmatrix}
$$

📌 Draw:
- First: 2 left, 2 down.
- Second: 3 right, 1 down.
- Resultant: 1 right, 3 down → (1, –3)

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#### 6.
$$
\begin{pmatrix} 2 \\ -3 \end{pmatrix} + \begin{pmatrix} -2 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ -2 \end{pmatrix}
$$

📌 Draw:
- First: 2 right, 3 down.
- Second: 2 left, 1 up.
- Resultant: 0 right, 2 down → (0, –2)

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✔ Section B Answers Summary:

| Q | Addition | Resultant |
|---|--------|----------|
| 2 | $\begin{pmatrix} 1 \\ 3 \end{pmatrix} + \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ | $\begin{pmatrix} 2 \\ 4 \end{pmatrix}$ |
| 3 | $\begin{pmatrix} 0 \\ 2 \end{pmatrix} + \begin{pmatrix} 2 \\ 2 \end{pmatrix}$ | $\begin{pmatrix} 2 \\ 4 \end{pmatrix}$ |
| 4 | $\begin{pmatrix} 4 \\ 1 \end{pmatrix} + \begin{pmatrix} -3 \\ 2 \end{pmatrix}$ | $\begin{pmatrix} 1 \\ 3 \end{pmatrix}$ |
| 5 | $\begin{pmatrix} -2 \\ -2 \end{pmatrix} + \begin{pmatrix} 3 \\ -1 \end{pmatrix}$ | $\begin{pmatrix} 1 \\ -3 \end{pmatrix}$ |
| 6 | $\begin{pmatrix} 2 \\ -3 \end{pmatrix} + \begin{pmatrix} -2 \\ 1 \end{pmatrix}$ | $\begin{pmatrix} 0 \\ -2 \end{pmatrix}$ |

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🎯 Final Notes:

- Section A: Use the grid to count horizontal and vertical movements.
- Section B: Add components algebraically, then sketch the vectors head-to-tail.
- Always remember: vector addition is commutative — order doesn’t matter, but direction and magnitude do.

Let me know if you’d like a visual sketch of any of the diagrams!