Vectors (Edexcel GCSE Maths): Revision Notes

Vectors

What are vectors?

Vectors represent movement with both a specific size and direction. Think of them as instructions that tell you how far to go and which way to travel. While they might seem a bit unusual at first, understanding a few key concepts will help you master vector problems.

Vector notation and representation

There are several different ways to write and represent vectors, and it's important to recognise all of them:

Column vectors

These show movement using coordinates in brackets. For example, the column vector $\begin{pmatrix} 2 \\ -5 \end{pmatrix}$ means move 2 units to the right and 5 units down. The top number shows horizontal movement (positive = right, negative = left) and the bottom number shows vertical movement (positive = up, negative = down).

Bold and underlined notation

In exam questions, vectors are often written in bold text like a or b. When you're writing by hand, you should underline vector letters like $\underline{a}$ or use arrows above them like $\vec{a}$. This helps distinguish vectors from ordinary numbers or measurements.

Vector from point to point

The notation $\overrightarrow{AB}$ means "the vector from point A to point B". This shows both the direction (A to B) and represents the movement needed to get from the first point to the second.

Multiplying vectors by scalars

When you multiply a vector by a positive number, you change the vector's size but keep its direction the same. This process is called scalar multiplication, and it literally scales the vector up or down.

Vectors that are scalar multiples of each other are always parallel to each other.

Adding and subtracting vectors

Vector addition and subtraction allow you to describe movements between points using vectors you already know. This is the foundation of most vector exam questions.

Vector addition

When you see "$\vec{a} + \vec{b}$", this means "go along vector a, then go along vector b". You can think of this as following a route - first follow one vector, then follow the next.

Vector subtraction

The expression "$\vec{a} - \vec{d}$" means "go along vector a, then go backwards along vector d". The minus sign tells you to travel in the opposite direction to vector d.

The route method

To find an unknown vector, you can "get there" by following any route made up of known vectors. This is incredibly useful for solving complex problems.

Vectors along a straight line

One of the most important applications of vectors is proving that points lie on a straight line. This connects vector algebra with geometric proof.

To prove three points are collinear (lie on the same straight line):

Vector ratios and proportions

Ratios become very useful in vector problems when you need to find vectors to points that divide lines in specific proportions.

When a point divides a line segment in a given ratio, you can use this information to find vector expressions. For example, if point E divides line DC in the ratio 3:1, then vector DE equals 3/4 of vector DC.

This technique is particularly powerful when working with parallelograms and other geometric shapes, where you can use the properties of parallel sides alongside ratio information.

Using vectors in geometric proofs

Vectors provide a powerful method for proving geometric properties. You can use vector methods to prove that shapes are parallelograms, that lines are parallel, or that points are collinear.

The general approach involves expressing unknown vectors in terms of known vectors, using properties like parallel sides or midpoints, applying vector operations to reach your conclusion, and interpreting the mathematical result geometrically.

Remember!