Column Vectors and Magnitude

A vector has a size and a direction. Write it as a column $\dbinom xy$ : $x$ across, $y$ up. Add, subtract and scale vectors component by component, and find the length (magnitude) with Pythagoras.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 25 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Read and write column vectors.
Add, subtract and multiply vectors by a number.
Find the vector $\overrightarrow{AB}$ between two points.
Find the magnitude $|\mathbf v|$ of a vector.

Pause and think

A plane flies $300$ km east and then $400$ km north. How far is it from where it started?

Section 2 of 14

Core idea

Start hereInstructions for a move

"Go 3 right and 4 down" is a complete instruction for a journey, whatever the starting point. Written as a column, $\begin{pmatrix}3\\-4\end{pmatrix}$ , it's a vector: it has a size and a direction.

The big idea

$\begin{pmatrix}x\\y\end{pmatrix}$ : $x$ across, $y$ up. Add, subtract, scale component by component. Magnitude $\sqrt{x^2+y^2}$ .

1. Arithmetic

Adding vectors: a + b, head to tail

\mathbf a

then

\mathbf b

; the direct route is

\mathbf a+\mathbf b

. Its length is

\sqrt{50}=7.07

Worked example $\mathbf a=\begin{pmatrix}3\\-4\end{pmatrix}$ , $\mathbf b=\begin{pmatrix}1\\2\end{pmatrix}$

$2\mathbf a-\mathbf b$ ; $|\mathbf a|$ ; $k$ if $\mathbf a+k\mathbf b$ has $x$ -component 7.

Goal: Three answers.
Strategy: Component by component.

$\begin{pmatrix}6-1\\-8-2\end{pmatrix}=\begin{pmatrix}5\\-10\end{pmatrix}$
Scale then subtract.
$\sqrt{9+16}=5$
Pythagoras.
$3+k=7\Rightarrow k=4$
Top components.

Answer

\begin{pmatrix}5\\-10\end{pmatrix}

;

5

;

4

✓ Check

x

2(3)-1=5

;

y

2(-4)-2=-10

✓.

Common misconceptionMagnitude with negatives Square first:

(-4)^2=16

, so length is never negative.

Pause and think

Are $\begin{pmatrix}2\\3\end{pmatrix}$ and $\begin{pmatrix}6\\9\end{pmatrix}$ parallel?

Summary

Component-wise arithmetic.
Magnitude by Pythagoras.

Magnitude: Length of a vector.

HistoryArrows for forces. Simon Stevin, a Flemish engineer, showed around 1586 that forces could be combined by drawing them as arrows end to end, an early version of vector addition. The column notation used in schools today came much later, in the 20th century.

Fun factWind and aircraft. A plane's ground speed is the vector sum of its airspeed and the wind. A strong tailwind can let a transatlantic flight from New York to London arrive an hour early, while the return trip against the jet stream takes longer.

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