Vector addition examples on a grid with column vector answers.

Let’s solve each problem step by step.

First, we need to find the column vector for each letter (a, b, c, d) by looking at how many units they move right/left and up/down on the grid.

Remember:
→ Right = positive x-direction
→ Left = negative x-direction
→ Up = positive y-direction
→ Down = negative y-direction

We write vectors as:
[ horizontal change ]
[ vertical change ]

---

Step 1: Find vector a

Vector a starts at bottom-left and goes to top-right.
From start to end:
- Moves 3 units right → +3
- Moves 2 units up → +2
So, a = [3, 2]

---

Step 2: Find vector b

Vector b goes down and slightly right.
From start to end:
- Moves 1 unit right → +1
- Moves 4 units down → -4
So, b = [1, -4]

---

Step 3: Find vector c

Vector c is horizontal to the right.
- Moves 2 units right → +2
- No vertical movement → 0
So, c = [2, 0]

---

Step 4: Find vector d

Vector d goes down and left.
From start to end:
- Moves 2 units left → -2
- Moves 2 units down → -2
So, d = [-2, -2]

---

Now let’s compute each sum:

---

1) a + b = [3, 2] + [1, -4] = [3+1, 2+(-4)] = [4, -2]

2) b + a = [1, -4] + [3, 2] = [1+3, -4+2] = [4, -2]
(Same as #1 — addition is commutative!)

3) a + c = [3, 2] + [2, 0] = [3+2, 2+0] = [5, 2]

4) c + d = [2, 0] + [-2, -2] = [2+(-2), 0+(-2)] = [0, -2]

5) d + c = [-2, -2] + [2, 0] = [-2+2, -2+0] = [0, -2]
(Same as #4)

6) b + c = [1, -4] + [2, 0] = [1+2, -4+0] = [3, -4]

7) d + b + c = [-2, -2] + [1, -4] + [2, 0]

Add step by step:
First: [-2, -2] + [1, -4] = [-1, -6]
Then: [-1, -6] + [2, 0] = [1, -6]

8) a + b + c + d = [3, 2] + [1, -4] + [2, 0] + [-2, -2]

Add in order:
[3+1, 2+(-4)] = [4, -2]
[4+2, -2+0] = [6, -2]
[6+(-2), -2+(-2)] = [4, -4]

Alternatively, add all x-components: 3 + 1 + 2 + (-2) = 4
All y-components: 2 + (-4) + 0 + (-2) = -4
So [4, -4]

---

Final Answer:
1) [4, -2]
2) [4, -2]
3) [5, 2]
4) [0, -2]
5) [0, -2]
6) [3, -4]
7) [1, -6]
8) [4, -4]