Adding and Subtracting Vectors

Add and subtract column vectors, see the result as one move, find a missing vector, and tidy sums of a and b.

In the exam for Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, IGCSE Additional Maths 0606 · 29 real past-paper questions below

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What Adding Vectors Means

Adding two vectors means doing one move, then the other. The sum is the single move from the very start to the very end.

Rule

Add the top numbers. Add the bottom numbers. Never mix a top with a bottom.

Key idea

Order does not matter: $a + b = b + a$ . Both routes end at the same place.

Key idea

Pilots add vectors. If a plane flies east at 800 km/h and the wind pushes it south at 60 km/h, in one hour its real move over the ground is $(8000) + (0 - 60) = (800 - 60)$ km.

Adding Column Vectors

The most common question: work out $a + b$ as a column vector. Exams may call the answer the resultant.

Worked example

1
$a = (3 - 2), b = (- 1 5)$ — Two column vectors. Adding them gives one move that does both.
2
$3 + (- 1) = 2$ — Add the tops (the across numbers). Adding −1 takes 1 away.
3
$- 2 + 5 = 3$ — Add the bottoms (the up or down numbers). 5 up cancels 2 down and leaves 3 up.
4
$a + b = (23)$ — Put both answers in one column. This single move does a then b.

Common mistake

Treating 3 + (−1) as 4. Adding a negative number makes it smaller.

Subtracting Vectors

Take the top from the top and the bottom from the bottom. As a picture, $a - b$ means do $a$ , then do $b$ backwards.

Worked example

1
$a - b = (3 - 2) - (- 1 5)$ — Same a and b as before. Subtract b from a, top from top and bottom from bottom.
2
$3 - (- 1) = 3 + 1 = 4$ — Taking away a negative is the same as adding, so a minus before −1 turns into a plus.
3
$- 2 - 5 = - 7$ — Take 5 away from −2. It goes further below zero.
4
$a - b = (4 - 7)$ — Put both answers in one column.

Common mistake

Getting 2 for the top. Taking away a negative adds: $3 - (- 1) = 3 + 1 = 4$ .

Key idea

To draw $a - b$ on a grid, draw $a$ , then from its tip draw $b$ backwards. The arrow from the start to the final tip is $a - b$ .

Finding a Missing Vector

Some questions hide letters inside the vectors. Work one row at a time: top row first, then bottom row.

Worked example

1
$(p 4) + (3 q) = (- 2 1)$ — Letters hide inside the vectors. Each row of the answer comes from the same row on the left.
2
$p + 3 = - 2$ — Write the top row as an equation: the two tops add to −2.
3
$p = - 2 - 3 = - 5$ — Take 3 away from both sides to get p alone.
4
$4 + q = 1$ — Write the bottom row as an equation: the two bottoms add to 1.
5
$q = 1 - 4 = - 3$ — Take 4 away from both sides to get q alone.

The same idea works for subtraction. Be careful with the minus in front of the unknown.

Worked example

1
$(m 5) - (2 n) = (7 - 1)$ — Subtract row by row, then write each row as an equation.
2
$m - 2 = 7$ — The top row: m minus 2 equals 7.
3
$m = 7 + 2 = 9$ — Add 2 to both sides to get m alone.
4
$5 - n = - 1$ — The bottom row: 5 minus n equals −1.
5
$- n = - 1 - 5 = - 6$ — Take 5 from both sides. This leaves minus n.
6
$n = 6$ — Change the sign on both sides, because we want n, not −n.

Key idea

Check by putting the numbers back: $- 5 + 3 = - 2$ and $4 + (- 3) = 1$ .

Adding Vectors Written With Letters

The 2-mark question in a shape gives an answer like $4 b - 2 a$ . Collect the $a$ terms and the $b$ terms, like algebra.

Worked example

1
$(2 a + 3 b) + (a - 5 b)$ — Add the two brackets. Collect the a terms together and the b terms together.
2
$= 2 a + 3 b + a - 5 b$ — Adding changes no signs, so just remove the brackets.
3
$2 a + a = 3 a$ — 2 lots of a plus 1 lot of a is 3 lots of a.
4
$3 b - 5 b = - 2 b$ — 3 lots of b take away 5 lots of b leaves −2 lots of b.
5
$= 3 a - 2 b$ — Write the two collected terms together.

Subtracting brackets needs more care.

Worked example

1
$(4 a - b) - (a - 3 b)$ — Subtract the second bracket from the first.
2
$= 4 a - b - a + 3 b$ — The minus outside flips every sign inside: −(a) is −a, and −(−3b) is +3b.
3
$4 a - a = 3 a$ — 4 lots of a take away 1 lot of a is 3 lots of a.
4
$- b + 3 b = 2 b$ — −1 lot of b plus 3 lots of b is 2 lots of b.
5
$= 3 a + 2 b$ — Write the two collected terms together.

Common mistake

Only flipping the first sign in the bracket. The minus sign applies to every term inside.

Which Method? Read the Question

Look for columns or letters, plus or minus.

You see	You do
a + b with columns	add tops, add bottoms
a − b	top minus top, bottom minus bottom
Letters inside a column	one row at a time
Brackets of a and b	collect like terms
A number times a vector	Multiplying a Vector by a Scalar
A route through a shape	Expressing Vectors in Terms of a and b
Read a vector from a grid	Column Vectors and Translations

Key idea

Draw a quick sketch, nose to tail. If your answer points somewhere odd, check the signs.

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