G25a - Adding and subtracting column vectors - BossMaths.com - Free Printable

Let’s solve each problem step by step. First, we need to find the column vector for each arrow (a, b, c, d) by counting how many units right/left and up/down they go.

Remember:
→ Right is positive x, Left is negative x
→ Up is positive y, Down is negative y
Column vector = [x, y]ᵀ → written as a vertical pair:
  [ x ]
  [ y ]

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Step 1: Find vectors a, b, c, d

Look at vector a:
Starts at bottom-left of its grid square, goes 3 right and 2 up.
So, a = [3, 2]

Vector b:
Goes 1 right and 4 down.
So, b = [1, -4]

Vector c:
Goes 2 right and 0 up/down.
So, c = [2, 0]

Vector d:
Goes 2 left and 2 down? Wait — let’s check direction.
Arrow points from top-right to bottom-left: so it goes 2 left and 2 down.
But in vectors, we care about displacement from start to end.
From tail to head: moves 2 units left (so x = -2), and 2 units down (y = -2).
Wait — actually, looking again: if you trace from start to end, it goes 2 left and 2 down? Let me count carefully.

Actually, let’s recheck all with precise grid counting:

Assume each grid line is 1 unit.

For a:
Tail at (say) (0,0), head at (3,2) → Δx=3, Δy=2 → [3, 2]

For b:
Tail at (say) (1,3), head at (2,-1) → Δx=1, Δy=-4 → [1, -4]

For c:
Tail at (6,1), head at (8,1) → Δx=2, Δy=0 → [2, 0]

For d:
Tail at (7,5), head at (5,3) → Δx = 5-7 = -2, Δy = 3-5 = -2 → [-2, -2]

Yes! So:

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

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Now compute each sum:

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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 above (addition is commutative)

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

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

5) d + c
[-2, -2] + [2, 0] = [0, -2] → same as above

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

7) d + b + c
First, d + b = [-2, -2] + [1, -4] = [-1, -6]
Then add c: [-1, -6] + [2, 0] = [1, -6]

Or directly:
d + b + c = [-2+1+2, -2+(-4)+0] = [1, -6]

8) a + b + c + d
Add all x-components: 3 + 1 + 2 + (-2) = 4
Add all y-components: 2 + (-4) + 0 + (-2) = -4
So [4, -4]

Double-check:
a+b = [4,-2], c+d = [0,-2], then [4,-2] + [0,-2] = [4,-4] → correct.

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Final Answer:

1) $\begin{bmatrix} 4 \\ -2 \end{bmatrix}$
2) $\begin{bmatrix} 4 \\ -2 \end{bmatrix}$
3) $\begin{bmatrix} 5 \\ 2 \end{bmatrix}$
4) $\begin{bmatrix} 0 \\ -2 \end{bmatrix}$
5) $\begin{bmatrix} 0 \\ -2 \end{bmatrix}$
6) $\begin{bmatrix} 3 \\ -4 \end{bmatrix}$
7) $\begin{bmatrix} 1 \\ -6 \end{bmatrix}$
8) $\begin{bmatrix} 4 \\ -4 \end{bmatrix}$