This worksheet helps students practice adding column vectors through visual diagrams and written calculations.

Problem Overview:

The task involves solving vector addition problems in two sections:
1. Section A: Given diagrams of vectors, write the resultant column vector.
2. Section B: Given column vectors, draw the resultant vector.

Solution Approach:

#### Section A: Diagrams to Resultant Column Vectors
To solve this section, we need to analyze each diagram and determine the horizontal (x-component) and vertical (y-component) displacements of the resultant vector. The resultant column vector will be written as:
\[
\begin{pmatrix}
x \\
y
\end{pmatrix}
\]
where \( x \) is the total horizontal displacement and \( y \) is the total vertical displacement.

##### Step-by-Step Analysis for Each Diagram:

1. Diagram 1 (Example):
- Horizontal displacement: \( +3 \)
- Vertical displacement: \( +2 \)
- Resultant vector:
\[
\begin{pmatrix}
3 \\
2
\end{pmatrix}
\]

2. Diagram 2:
- Horizontal displacement: \( +1 \)
- Vertical displacement: \( +4 \)
- Resultant vector:
\[
\begin{pmatrix}
1 \\
4
\end{pmatrix}
\]

3. Diagram 3:
- Horizontal displacement: \( +2 \)
- Vertical displacement: \( +1 \)
- Resultant vector:
\[
\begin{pmatrix}
2 \\
1
\end{pmatrix}
\]

4. Diagram 4:
- Horizontal displacement: \( +3 \)
- Vertical displacement: \( +1 \)
- Resultant vector:
\[
\begin{pmatrix}
3 \\
1
\end{pmatrix}
\]

5. Diagram 5:
- Horizontal displacement: \( +2 \)
- Vertical displacement: \( -1 \)
- Resultant vector:
\[
\begin{pmatrix}
2 \\
-1
\end{pmatrix}
\]

6. Diagram 6:
- Horizontal displacement: \( +2 \)
- Vertical displacement: \( +3 \)
- Resultant vector:
\[
\begin{pmatrix}
2 \\
3
\end{pmatrix}
\]

7. Diagram 7:
- Horizontal displacement: \( +2 \)
- Vertical displacement: \( -2 \)
- Resultant vector:
\[
\begin{pmatrix}
2 \\
-2
\end{pmatrix}
\]

8. Diagram 8:
- Horizontal displacement: \( -4 \)
- Vertical displacement: \( -1 \)
- Resultant vector:
\[
\begin{pmatrix}
-4 \\
-1
\end{pmatrix}
\]

9. Diagram 9:
- Horizontal displacement: \( +3 \)
- Vertical displacement: \( -2 \)
- Resultant vector:
\[
\begin{pmatrix}
3 \\
-2
\end{pmatrix}
\]

#### Section B: Column Vectors to Resultant Diagrams
To solve this section, we need to add the given column vectors component-wise:
\[
\begin{pmatrix}
a_1 \\
b_1
\end{pmatrix}
+
\begin{pmatrix}
a_2 \\
b_2
\end{pmatrix}
=
\begin{pmatrix}
a_1 + a_2 \\
b_1 + b_2
\end{pmatrix}
\]
Then, we draw the resultant vector based on the calculated components.

##### Step-by-Step Calculation for Each Pair:

1. Pair 1 (Example):
\[
\begin{pmatrix}
3 \\
1
\end{pmatrix}
+
\begin{pmatrix}
0 \\
2
\end{pmatrix}
=
\begin{pmatrix}
3 + 0 \\
1 + 2
\end{pmatrix}
=
\begin{pmatrix}
3 \\
3
\end{pmatrix}
\]
- Draw a vector with horizontal displacement \( +3 \) and vertical displacement \( +3 \).

2. Pair 2:
\[
\begin{pmatrix}
1 \\
3
\end{pmatrix}
+
\begin{pmatrix}
1 \\
1
\end{pmatrix}
=
\begin{pmatrix}
1 + 1 \\
3 + 1
\end{pmatrix}
=
\begin{pmatrix}
2 \\
4
\end{pmatrix}
\]
- Draw a vector with horizontal displacement \( +2 \) and vertical displacement \( +4 \).

3. Pair 3:
\[
\begin{pmatrix}
0 \\
2
\end{pmatrix}
+
\begin{pmatrix}
2 \\
2
\end{pmatrix}
=
\begin{pmatrix}
0 + 2 \\
2 + 2
\end{pmatrix}
=
\begin{pmatrix}
2 \\
4
\end{pmatrix}
\]
- Draw a vector with horizontal displacement \( +2 \) and vertical displacement \( +4 \).

4. Pair 4:
\[
\begin{pmatrix}
4 \\
1
\end{pmatrix}
+
\begin{pmatrix}
-3 \\
2
\end{pmatrix}
=
\begin{pmatrix}
4 + (-3) \\
1 + 2
\end{pmatrix}
=
\begin{pmatrix}
1 \\
3
\end{pmatrix}
\]
- Draw a vector with horizontal displacement \( +1 \) and vertical displacement \( +3 \).

5. Pair 5:
\[
\begin{pmatrix}
-2 \\
-2
\end{pmatrix}
+
\begin{pmatrix}
3 \\
-1
\end{pmatrix}
=
\begin{pmatrix}
-2 + 3 \\
-2 + (-1)
\end{pmatrix}
=
\begin{pmatrix}
1 \\
-3
\end{pmatrix}
\]
- Draw a vector with horizontal displacement \( +1 \) and vertical displacement \( -3 \).

6. Pair 6:
\[
\begin{pmatrix}
2 \\
-3
\end{pmatrix}
+
\begin{pmatrix}
-2 \\
1
\end{pmatrix}
=
\begin{pmatrix}
2 + (-2) \\
-3 + 1
\end{pmatrix}
=
\begin{pmatrix}
0 \\
-2
\end{pmatrix}
\]
- Draw a vector with horizontal displacement \( 0 \) and vertical displacement \( -2 \).

Final Answers:

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

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

Boxed Final Answer:

\[
\boxed{
\text{Section A: }
\begin{pmatrix} 3 \\ 2 \end{pmatrix},
\begin{pmatrix} 1 \\ 4 \end{pmatrix},
\begin{pmatrix} 2 \\ 1 \end{pmatrix},
\begin{pmatrix} 3 \\ 1 \end{pmatrix},
\begin{pmatrix} 2 \\ -1 \end{pmatrix},
\begin{pmatrix} 2 \\ 3 \end{pmatrix},
\begin{pmatrix} 2 \\ -2 \end{pmatrix},
\begin{pmatrix} -4 \\ -1 \end{pmatrix},
\begin{pmatrix} 3 \\ -2 \end{pmatrix}
}
\]
\[
\boxed{
\text{Section B: }
\begin{pmatrix} 3 \\ 3 \end{pmatrix},
\begin{pmatrix} 2 \\ 4 \end{pmatrix},
\begin{pmatrix} 2 \\ 4 \end{pmatrix},
\begin{pmatrix} 1 \\ 3 \end{pmatrix},
\begin{pmatrix} 1 \\ -3 \end{pmatrix},
\begin{pmatrix} 0 \\ -2 \end{pmatrix}
}
\]