Vectors in maths: column vectors, magnitude and proofs

TutorAura guide · Updated 9 November 2026

A vector has both size (magnitude) and direction, unlike a scalar, which only has size. In maths, vectors describe movements: "3 right and 4 up". They're written as column vectors, bold letters such as $\mathbf{a}$ , or with arrows such as $\overrightarrow{AB}$ . GCSE and IGCSE questions range from simple arithmetic to geometric proofs.

Key takeaways

A column vector shows the move across (top) and up or down (bottom).
Add and subtract vectors one component at a time.
Magnitude uses Pythagoras: the length of (3, 4) is 5.
Parallel vectors are multiples of each other.

Column vectors

\begin{pmatrix} 3 \\ -2 \end{pmatrix}

means 3 to the right and 2 down. The top number is the horizontal move (positive right, negative left); the bottom is the vertical move (positive up, negative down). It's the same notation used for translations in transformations.

Adding, subtracting and multiplying by a number

Let $\mathbf{a} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -3 \\ 4 \end{pmatrix}$ .

$\mathbf{a} + \mathbf{b} = \begin{pmatrix} -1 \\ 3 \end{pmatrix}$ (add the tops, add the bottoms).
$2\mathbf{a} = \begin{pmatrix} 4 \\ -2 \end{pmatrix}$ (multiply both numbers).
$2\mathbf{a} - \mathbf{b} = \begin{pmatrix} 4 - (-3) \\ -2 - 4 \end{pmatrix} = \begin{pmatrix} 7 \\ -6 \end{pmatrix}$ .

$-\mathbf{a}$ is the same length as $\mathbf{a}$ but points the opposite way.

Magnitude

The magnitude (length) of a vector comes from Pythagoras:

\left| \begin{pmatrix} x \\ y \end{pmatrix} \right| = \sqrt{x^2 + y^2}

The magnitude of $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ is $\sqrt{9 + 16} = 5$ . Check it with our Pythagoras calculator.

Vectors between points

If A is at $(1, 2)$ and B is at $(5, -1)$ , then

\overrightarrow{AB} = \begin{pmatrix} 5 - 1 \\ -1 - 2 \end{pmatrix} = \begin{pmatrix} 4 \\ -3 \end{pmatrix}

and $|\overrightarrow{AB}| = 5$ , the same answer as the distance between two points. With position vectors $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ : $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$ ("end minus start").

Vector journeys in geometry

To find a vector in a diagram, find a route along vectors you know. Going against an arrow means subtracting.

Example: $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ . M is the midpoint of AB.

\overrightarrow{OM} = \overrightarrow{OA} + \tfrac{1}{2}\overrightarrow{AB} = \mathbf{a} + \tfrac{1}{2}(\mathbf{b} - \mathbf{a}) = \tfrac{1}{2}\mathbf{a} + \tfrac{1}{2}\mathbf{b}

If instead P divides AB in the ratio $1 : 2$ , then $\overrightarrow{AP} = \tfrac{1}{3}\overrightarrow{AB}$ , so $\overrightarrow{OP} = \mathbf{a} + \tfrac{1}{3}(\mathbf{b} - \mathbf{a}) = \tfrac{2}{3}\mathbf{a} + \tfrac{1}{3}\mathbf{b}$ . A ratio of $1 : 2$ means thirds, not halves.

Parallel and collinear

Two vectors are parallel if one is a multiple of the other: $\begin{pmatrix} 6 \\ -4 \end{pmatrix} = 2\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ , so they're parallel.
Three points are collinear (on one straight line) if, say, $\overrightarrow{AB}$ and $\overrightarrow{BC}$ are parallel and share the point B.

In a proof, write the conclusion in words: " $\overrightarrow{AC} = 3\overrightarrow{AB}$ , so AC is parallel to AB, and they share point A, so A, B and C are collinear."

Common mistakes

Swapping the top and bottom numbers.
Getting $\overrightarrow{AB}$ the wrong way round: it's $\mathbf{b} - \mathbf{a}$ , not $\mathbf{a} - \mathbf{b}$ .
Forgetting to simplify: collect the $\mathbf{a}$ terms and the $\mathbf{b}$ terms.
Leaving out the words in a proof.

Practice questions

$\mathbf{p} = \begin{pmatrix} 1 \\ 5 \end{pmatrix}$ , $\mathbf{q} = \begin{pmatrix} 4 \\ -2 \end{pmatrix}$ . Find $3\mathbf{p} + \mathbf{q}$ .
Find the magnitude of $\begin{pmatrix} -6 \\ 8 \end{pmatrix}$ .
Are $\begin{pmatrix} 2 \\ 5 \end{pmatrix}$ and $\begin{pmatrix} -4 \\ -10 \end{pmatrix}$ parallel?

Answers: 1. $\begin{pmatrix} 7 \\ 13 \end{pmatrix}$ . 2. $\sqrt{36 + 64} = 10$ . 3. Yes: the second is $-2$ times the first (parallel, opposite direction).

Frequently asked questions

What is a vector in maths?

A vector is a quantity with size and direction, such as a movement of 3 right and 4 up. It's often written as a column vector.

How do you find the magnitude of a vector?

Use Pythagoras: square each component, add them and take the square root. The magnitude of (3, 4) is 5.

How do you show two vectors are parallel?

Show that one is a multiple of the other, for example (6, 9) = 3 × (2, 3).

Sources

GCSE mathematics: subject content and assessment objectives. Department for Education.
Cambridge IGCSE Mathematics 0580 syllabus. Cambridge International.

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