Examples of vector addition using column vectors on a grid.

Let's solve this step by step.

We are given a grid with four vectors: a, b, c, and d. We need to determine their components (as column vectors) by counting the horizontal (x) and vertical (y) movements from the start to the end of each vector.

---

Step 1: Determine the vectors a, b, c, and d

We'll use the grid to find the displacement in x and y directions.

Assume each square is 1 unit.

#### Vector a:
- Starts at approximately (-4, 0)
- Ends at (0, 3)
- So, change in x: 0 - (-4) = 4
- Change in y: 3 - 0 = 3
- So, a =
$$
\begin{bmatrix}
4 \\
3
\end{bmatrix}
$$

#### Vector b:
- Starts at (-2, 2)
- Ends at (0, -2)
- Change in x: 0 - (-2) = 2
- Change in y: -2 - 2 = -4
- So, b =
$$
\begin{bmatrix}
2 \\
-4
\end{bmatrix}
$$

#### Vector c:
- Starts at (5, 0)
- Ends at (7, 0)
- Change in x: 7 - 5 = 2
- Change in y: 0 - 0 = 0
- So, c =
$$
\begin{bmatrix}
2 \\
0
\end{bmatrix}
$$

#### Vector d:
- Starts at (6, 3)
- Ends at (9, 0)
- Change in x: 9 - 6 = 3
- Change in y: 0 - 3 = -3
- So, d =
$$
\begin{bmatrix}
3 \\
-3
\end{bmatrix}
$$

---

Now we can compute each sum as requested.

---

Now solve each:

1) a + b
$$
\begin{bmatrix}
4 \\
3
\end{bmatrix}
+
\begin{bmatrix}
2 \\
-4
\end{bmatrix}
=
\begin{bmatrix}
6 \\
-1
\end{bmatrix}
$$

2) b + a
Same as above since addition is commutative:
$$
\begin{bmatrix}
2 \\
-4
\end{bmatrix}
+
\begin{bmatrix}
4 \\
3
\end{bmatrix}
=
\begin{bmatrix}
6 \\
-1
\end{bmatrix}
$$

3) a + c
$$
\begin{bmatrix}
4 \\
3
\end{bmatrix}
+
\begin{bmatrix}
2 \\
0
\end{bmatrix}
=
\begin{bmatrix}
6 \\
3
\end{bmatrix}
$$

4) c + d
$$
\begin{bmatrix}
2 \\
0
\end{bmatrix}
+
\begin{bmatrix}
3 \\
-3
\end{bmatrix}
=
\begin{bmatrix}
5 \\
-3
\end{bmatrix}
$$

5) d + c
Same as above:
$$
\begin{bmatrix}
3 \\
-3
\end{bmatrix}
+
\begin{bmatrix}
2 \\
0
\end{bmatrix}
=
\begin{bmatrix}
5 \\
-3
\end{bmatrix}
$$

6) b + c
$$
\begin{bmatrix}
2 \\
-4
\end{bmatrix}
+
\begin{bmatrix}
2 \\
0
\end{bmatrix}
=
\begin{bmatrix}
4 \\
-4
\end{bmatrix}
$$

7) d + b + c
Add them step by step:

First, d + b:
$$
\begin{bmatrix}
3 \\
-3
\end{bmatrix}
+
\begin{bmatrix}
2 \\
-4
\end{bmatrix}
=
\begin{bmatrix}
5 \\
-7
\end{bmatrix}
$$

Then add c:
$$
\begin{bmatrix}
5 \\
-7
\end{bmatrix}
+
\begin{bmatrix}
2 \\
0
\end{bmatrix}
=
\begin{bmatrix}
7 \\
-7
\end{bmatrix}
$$

8) a + b + c + d
We can add all together:

$$
\begin{bmatrix}
4 \\
3
\end{bmatrix}
+
\begin{bmatrix}
2 \\
-4
\end{bmatrix}
+
\begin{bmatrix}
2 \\
0
\end{bmatrix}
+
\begin{bmatrix}
3 \\
-3
\end{bmatrix}
=
\begin{bmatrix}
4+2+2+3 \\
3-4+0-3
\end{bmatrix}
=
\begin{bmatrix}
11 \\
-4
\end{bmatrix}
$$

---

Final Answers (as column vectors):

1)
$$
\begin{bmatrix}
6 \\
-1
\end{bmatrix}
$$

2)
$$
\begin{bmatrix}
6 \\
-1
\end{bmatrix}
$$

3)
$$
\begin{bmatrix}
6 \\
3
\end{bmatrix}
$$

4)
$$
\begin{bmatrix}
5 \\
-3
\end{bmatrix}
$$

5)
$$
\begin{bmatrix}
5 \\
-3
\end{bmatrix}
$$

6)
$$
\begin{bmatrix}
4 \\
-4
\end{bmatrix}
$$

7)
$$
\begin{bmatrix}
7 \\
-7
\end{bmatrix}
$$

8)
$$
\begin{bmatrix}
11 \\
-4
\end{bmatrix}
$$

✔ All answers are now in column vector form as requested.