IGCSEMathsCambridge (CIE)ExtendedRevision NotesVectors & TransformationsVectorsIntroduction to Column Vectors

Introduction to Column Vectors (Cambridge (CIE) IGCSE Maths: Extended): Revision Note

Exam code: 0580 & 0980

Written by: Jamie Wood

Reviewed by: Dan Finlay

Updated on 19 January 2026

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Basic vectors

What are column vectors?

A column vector can be used to describe how to get from one point to another point
- This is also called a translation vector
- $(\begin{matrix} 6 \\ 3 \end{matrix})$ means 6 units to the right and 3 units up

How do I add and subtract column vectors?

Adding and subtracting vectors is done by looking at the top numbers and bottom numbers separately
To add column vectors
- Add the top numbers together
- Add the bottom numbers together
  - $(\begin{matrix} 5 \\ 2 \end{matrix}) + (\begin{matrix} 3 \\ - 1 \end{matrix}) = (\begin{matrix} 5 + 3 \\ 2 + (- 1) \end{matrix}) = (\begin{matrix} 8 \\ 1 \end{matrix})$
To subtract column vectors
- Subtract the second top number from the first
- Subtract the second bottom number from the first
  - $(\begin{matrix} 5 \\ 2 \end{matrix}) - (\begin{matrix} 3 \\ - 1 \end{matrix}) = (\begin{matrix} 5 - 3 \\ 2 - (- 1) \end{matrix}) = (\begin{matrix} 2 \\ 3 \end{matrix})$

How do I multiply a vector by a scalar?

A scalar is a number not a vector
- It does not have a direction
To multiply a column vector by a scalar
- Multiply the top number by the scalar
- Multiply the bottom number by the scalar
  - $3 (\begin{matrix} 2 \\ - 1 \end{matrix}) = (\begin{matrix} 3 \times 2 \\ 3 \times (- 1) \end{matrix}) = (\begin{matrix} 6 \\ - 3 \end{matrix})$

How do I write an expression as a single column vector?

You need to follow the order of operations
- $2 (\begin{matrix} 5 \\ 2 \end{matrix}) + 5 (\begin{matrix} 3 \\ - 1 \end{matrix})$
STEP 1
Multiply each vector by the scalar in front of it
- $(\begin{matrix} 2 \times 5 \\ 2 \times 2 \end{matrix}) + (\begin{matrix} 5 \times 3 \\ 5 \times (- 1) \end{matrix}) = (\begin{matrix} 10 \\ 4 \end{matrix}) + (\begin{matrix} 15 \\ - 5 \end{matrix})$
STEP 2
Add or subtract the new column vectors
- $(\begin{matrix} 10 + 15 \\ 4 + (- 5) \end{matrix}) = (\begin{matrix} 25 \\ - 1 \end{matrix})$

Worked Example

$a = (\begin{matrix} p \\ 3 \end{matrix})$ and $b = (\begin{matrix} - 2 \\ 1 \end{matrix})$ .

Given that $2 a + 3 b = (\begin{matrix} 4 \\ q \end{matrix})$ , find the value of $p$ and the value of $q$ .

Answer:

Write the left-side side as one vector
Multiply each vector by the scalar in front of it

$(\begin{matrix} 2 p \\ 6 \end{matrix}) + (\begin{matrix} - 6 \\ 3 \end{matrix}) = (\begin{matrix} 4 \\ q \end{matrix})$

Add the vectors together

$(\begin{matrix} 2 p - 6 \\ 9 \end{matrix}) = (\begin{matrix} 4 \\ q \end{matrix})$

The top components are equal
Form and solve an equation

$\begin{array}{rcl} 2 p - 6 & = & 4 \\ 2 p & = & 10 \\ p & = & 5 \end{array}$

The bottom components are equal

$9 = q$

$p = 5$ and $q = 9$

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Questions students ask about Introduction to Column Vectors

Real questions from students revising this topic on Save My Exams

How do you solve algebraic column vector questions?

Write each side as a single column vector, then equate the top components and the bottom components separately to get equations in your unknowns.

Multiply each vector by the scalar in front of it — the scalar multiplies both components, letters included.
Add or subtract the results, top with top and bottom with bottom, so there is one column vector on each side.
Form one equation from the top numbers and a separate equation from the bottom numbers.
Solve each one.

Using the example on this page, and writing a column vector as (top, bottom), a = (p, 3), b = (−2, 1) and 2a + 3b = (4, q):

multiply by the scalars: (2p, 6) + (−6, 3) = (4, q)
add the vectors: (2p − 6, 9) = (4, q)
top components: 2p − 6 = 4, so 2p = 10 and p = 5
bottom components: 9 = q

So p = 5 and q = 9. Keep the top and bottom rows completely separate throughout — never mix a top number with a bottom one.

How do you know which way to subtract coordinates when finding a column vector?

Always do end point minus start point: the top number is (x of end − x of start), the bottom number is (y of end − y of start).

For the vector AB you start at A and finish at B, so subtract A's coordinates from B's.

Example: from (1, 2) to (7, 5).

Top: 7 − 1 = 6
Bottom: 5 − 2 = 3
The column vector has 6 on top and 3 underneath — 6 units right and 3 units up, which is the page's own example vector

Check with the signs from the page's diagram: + on top means right, − means left; + underneath means up, − means down. The end point here is to the right of and above the start, so both numbers should be positive.

Subtract the wrong way round and you get −6 and −3. That is the journey back, the same size but pointing the opposite way.

On a grid you can just count squares instead: across first, then up or down.