Column Vectors (A) worksheet for GCSE Geometry, covering vector representation and addition.

Problem Overview:

The task involves working with column vectors. In Section A, we need to write the given vectors as column vectors and draw specified column vectors. In Section B, we need to draw the sum of given vectors and write the sum of other vectors.

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Section A: Column Vectors

#### Questions 1-6: Write as column vectors
For each question, we need to determine the horizontal (x) and vertical (y) components of the vector and write them as a column vector.

1. Question 1:
- The vector starts at \((1, 1)\) and ends at \((4, 4)\).
- Horizontal component: \(4 - 1 = 3\)
- Vertical component: \(4 - 1 = 3\)
- Column vector: \(\begin{pmatrix} 3 \\ 3 \end{pmatrix}\)

2. Question 2:
- The vector starts at \((1, 1)\) and ends at \((3, 0)\).
- Horizontal component: \(3 - 1 = 2\)
- Vertical component: \(0 - 1 = -1\)
- Column vector: \(\begin{pmatrix} 2 \\ -1 \end{pmatrix}\)

3. Question 3:
- The vector starts at \((1, 1)\) and ends at \((3, 2)\).
- Horizontal component: \(3 - 1 = 2\)
- Vertical component: \(2 - 1 = 1\)
- Column vector: \(\begin{pmatrix} 2 \\ 1 \end{pmatrix}\)

4. Question 4:
- The vector starts at \((1, 1)\) and ends at \((5, 1)\).
- Horizontal component: \(5 - 1 = 4\)
- Vertical component: \(1 - 1 = 0\)
- Column vector: \(\begin{pmatrix} 4 \\ 0 \end{pmatrix}\)

5. Question 5:
- The vector starts at \((1, 1)\) and ends at \((4, 3)\).
- Horizontal component: \(4 - 1 = 3\)
- Vertical component: \(3 - 1 = 2\)
- Column vector: \(\begin{pmatrix} 3 \\ 2 \end{pmatrix}\)

6. Question 6:
- The vector starts at \((1, 1)\) and ends at \((1, 0)\).
- Horizontal component: \(1 - 1 = 0\)
- Vertical component: \(0 - 1 = -1\)
- Column vector: \(\begin{pmatrix} 0 \\ -1 \end{pmatrix}\)

#### Questions 7-9: Draw the column vector
For each question, we need to draw the vector starting from the origin \((1, 1)\).

7. Question 7: \(\begin{pmatrix} 2 \\ 4 \end{pmatrix}\)
- Start at \((1, 1)\).
- Move 2 units right and 4 units up.
- End point: \((3, 5)\).

8. Question 8: \(\begin{pmatrix} -6 \\ 0 \end{pmatrix}\)
- Start at \((1, 1)\).
- Move 6 units left and 0 units up/down.
- End point: \((-5, 1)\).

9. Question 9: \(\begin{pmatrix} -3 \\ -3 \end{pmatrix}\)
- Start at \((1, 1)\).
- Move 3 units left and 3 units down.
- End point: \((-2, -2)\).

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Section B: Sum of Vectors

#### Questions 1-5: Draw the sum of the vectors
For each question, we need to add the given vectors graphically.

1. Question 1: \(\begin{pmatrix} 3 \\ 1 \end{pmatrix} + \begin{pmatrix} 0 \\ 2 \end{pmatrix}\)
- Add the components: \(\begin{pmatrix} 3 + 0 \\ 1 + 2 \end{pmatrix} = \begin{pmatrix} 3 \\ 3 \end{pmatrix}\).
- Draw the resulting vector from \((1, 1)\) to \((4, 4)\).

2. Question 2: \(\begin{pmatrix} 1 \\ 2 \end{pmatrix} + \begin{pmatrix} -3 \\ -3 \end{pmatrix}\)
- Add the components: \(\begin{pmatrix} 1 + (-3) \\ 2 + (-3) \end{pmatrix} = \begin{pmatrix} -2 \\ -1 \end{pmatrix}\).
- Draw the resulting vector from \((1, 1)\) to \((-1, 0)\).

3. Question 3: \(\begin{pmatrix} 4 \\ 0 \end{pmatrix} + \begin{pmatrix} -5 \\ -1 \end{pmatrix}\)
- Add the components: \(\begin{pmatrix} 4 + (-5) \\ 0 + (-1) \end{pmatrix} = \begin{pmatrix} -1 \\ -1 \end{pmatrix}\).
- Draw the resulting vector from \((1, 1)\) to \((0, 0)\).

4. Question 4: \(\begin{pmatrix} -2 \\ -4 \end{pmatrix} + \begin{pmatrix} -3 \\ 3 \end{pmatrix}\)
- Add the components: \(\begin{pmatrix} -2 + (-3) \\ -4 + 3 \end{pmatrix} = \begin{pmatrix} -5 \\ -1 \end{pmatrix}\).
- Draw the resulting vector from \((1, 1)\) to \((-4, 0)\).

5. Question 5: \(\begin{pmatrix} 0 \\ -2 \end{pmatrix} + \begin{pmatrix} -2 \\ -1 \end{pmatrix}\)
- Add the components: \(\begin{pmatrix} 0 + (-2) \\ -2 + (-1) \end{pmatrix} = \begin{pmatrix} -2 \\ -3 \end{pmatrix}\).
- Draw the resulting vector from \((1, 1)\) to \((-1, -2)\).

#### Questions 6-9: Write the sum of the vectors
For each question, we need to add the given vectors algebraically.

6. Question 6:
- The vector starts at \((1, 1)\) and ends at \((4, 4)\).
- Components: \(\begin{pmatrix} 4 - 1 \\ 4 - 1 \end{pmatrix} = \begin{pmatrix} 3 \\ 3 \end{pmatrix}\).

7. Question 7:
- The vector starts at \((1, 1)\) and ends at \((3, 2)\).
- Components: \(\begin{pmatrix} 3 - 1 \\ 2 - 1 \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}\).

8. Question 8:
- The vector starts at \((1, 1)\) and ends at \((2, 3)\).
- Components: \(\begin{pmatrix} 2 - 1 \\ 3 - 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}\).

9. Question 9:
- The vector starts at \((1, 1)\) and ends at \((4, 2)\).
- Components: \(\begin{pmatrix} 4 - 1 \\ 2 - 1 \end{pmatrix} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}\).

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Final Answers:

Section A:
1. \(\begin{pmatrix} 3 \\ 3 \end{pmatrix}\)
2. \(\begin{pmatrix} 2 \\ -1 \end{pmatrix}\)
3. \(\begin{pmatrix} 2 \\ 1 \end{pmatrix}\)
4. \(\begin{pmatrix} 4 \\ 0 \end{pmatrix}\)
5. \(\begin{pmatrix} 3 \\ 2 \end{pmatrix}\)
6. \(\begin{pmatrix} 0 \\ -1 \end{pmatrix}\)

Section B:
1. \(\begin{pmatrix} 3 \\ 3 \end{pmatrix}\)
2. \(\begin{pmatrix} -2 \\ -1 \end{pmatrix}\)
3. \(\begin{pmatrix} -1 \\ -1 \end{pmatrix}\)
4. \(\begin{pmatrix} -5 \\ -1 \end{pmatrix}\)
5. \(\begin{pmatrix} -2 \\ -3 \end{pmatrix}\)
6. \(\begin{pmatrix} 3 \\ 3 \end{pmatrix}\)
7. \(\begin{pmatrix} 2 \\ 1 \end{pmatrix}\)
8. \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\)
9. \(\begin{pmatrix} 3 \\ 1 \end{pmatrix}\)

Boxed Final Answer:
\[
\boxed{
\begin{array}{l}
\text{Section A:} \\
1. \begin{pmatrix} 3 \\ 3 \end{pmatrix}, \quad 2. \begin{pmatrix} 2 \\ -1 \end{pmatrix}, \quad 3. \begin{pmatrix} 2 \\ 1 \end{pmatrix}, \quad 4. \begin{pmatrix} 4 \\ 0 \end{pmatrix}, \quad 5. \begin{pmatrix} 3 \\ 2 \end{pmatrix}, \quad 6. \begin{pmatrix} 0 \\ -1 \end{pmatrix} \\
\text{Section B:} \\
1. \begin{pmatrix} 3 \\ 3 \end{pmatrix}, \quad 2. \begin{pmatrix} -2 \\ -1 \end{pmatrix}, \quad 3. \begin{pmatrix} -1 \\ -1 \end{pmatrix}, \quad 4. \begin{pmatrix} -5 \\ -1 \end{pmatrix}, \quad 5. \begin{pmatrix} -2 \\ -3 \end{pmatrix}, \quad 6. \begin{pmatrix} 3 \\ 3 \end{pmatrix}, \quad 7. \begin{pmatrix} 2 \\ 1 \end{pmatrix}, \quad 8. \begin{pmatrix} 1 \\ 2 \end{pmatrix}, \quad 9. \begin{pmatrix} 3 \\ 1 \end{pmatrix}
\end{array}
}
\]