VectorsCambridge IGCSE Maths: Revision notes

Section 1

What is a vector?

A vector is a quantity with both magnitude (size) and direction. Vectors are written in several ways:

As a column vector, e.g. $\binom{x}{y}$ , describing a translation of $x$ units horizontally and $y$ units vertically
Using two capital letters with an arrow, $\overrightarrow{AB}$ , meaning the vector from point A to point B
In bold, e.g. $\mathbf{a}$ , in printed text (underlined by hand, e.g. $\underline{a}$ )

A vector can be represented visually as a directed line segment — an arrow showing both length and direction.

Key termsvectorcolumn vector

Section 2

How do you add, subtract and scale vectors?

Vectors can be combined algebraically:

Addition: add the corresponding components, e.g. $\binom{2}{3} + \binom{1}{-4} = \binom{3}{-1}$
Subtraction: subtract the corresponding components, e.g. $\binom{2}{3} - \binom{1}{-4} = \binom{1}{7}$
Scalar multiplication: multiply every component by the scalar, e.g. $3\binom{2}{-1} = \binom{6}{-3}$

Multiplying a vector by a scalar changes its magnitude (and reverses its direction if the scalar is negative) but keeps it parallel to the original vector.

Key termsscalar

Exam tip

Two vectors are parallel if one is a scalar multiple of the other, e.g. b = 2a means b is parallel to a and twice its length.

Section 3

How do you find the magnitude of a vector?

The magnitude of a vector is its length, calculated using Pythagoras' theorem:

For a vector $\binom{x}{y}$ , magnitude = $\sqrt{x^2 + y^2}$

This works because the horizontal and vertical components form the two shorter sides of a right-angled triangle, with the vector itself as the hypotenuse.

Key termsmagnitude

Example

For the vector (3, 4): magnitude = √(3² + 4²) = √(9 + 16) = √25 = 5.

Section 4

What are position vectors, and how are vectors expressed in terms of others?

A position vector describes the position of a point relative to a fixed origin, O. The position vector of point A is written $\overrightarrow{OA}$ .

Vectors between two points can be found using position vectors: $\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}$

More generally, any vector in a diagram can be expressed as a sum or difference of two given (coplanar) vectors, by tracing a route between the start and end points using the given vectors and their multiples.

Key termsposition vector

Exam tip

To express a vector in terms of two others, trace a path from the start point to the end point, adding vectors when travelling 'with' the arrow and subtracting when travelling 'against' it.

Section 5

How are vectors used to solve geometric problems?

Vectors can be used to prove geometric facts without measuring:

Parallel lines: two vectors are parallel if one is a scalar multiple of the other, e.g. $\overrightarrow{PQ} = k\overrightarrow{RS}$
Collinear points: three points A, B, C are collinear if $\overrightarrow{AB}$ and $\overrightarrow{BC}$ are parallel (one is a scalar multiple of the other) AND they share a common point (B)
Ratio and similarity problems: if a point divides a line in a given ratio, its position vector can be found using a weighted combination of the position vectors at each end

Key termscollinear

Common mistake

To prove points are collinear, showing the vectors are parallel is not enough on its own — you must also show they share a common point.

Must Know

A vector has both magnitude and direction; written as a column vector, $\overrightarrow{AB}$ , or bold $\mathbf{a}$
Add/subtract vectors component-wise; scalar multiplication scales the magnitude and keeps the vector parallel to the original
Magnitude of $\binom{x}{y}$ is $\sqrt{x^2+y^2}$ (Pythagoras)
Position vectors describe a point's location relative to the origin O
$\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}$
Vectors prove parallel lines (scalar multiples) and collinear points (parallel AND a shared point)

That's the notes covered.

Carry on to the next subtopic.