Vector Basics and Ratio

Vectors is one of those topics that look intimidating at first, but once you see what they really are, everything clicks. In this note, we will build up from the very basics: what a vector is, how we write it down, and how we add vectors together. Then, we'll use those ideas to solve the kind of geometry and ratio problems that show up in every GCSE paper. We'll work through everything together, step by step, so don't worry if it feels new right now.

So far, in your maths, coordinates have told you where a point is, a fixed position on a grid, like A vector is different: it describes a movement from one place to another. A vector has two things involved in it:

a direction (which way you go)

a magnitude (how far you go)

It doesn’t matter where you start; the same vector can be drawn anywhere on the grid.

We usually write a vector in column form. The top number tells us how far we move across (right is positive, left is negative), and the bottom number tells us how far we move up or down (up is positive, down is negative).

Column Vectors

A column vector written

means: move across, then up.

A positive top number is right; negative is left.

A positive bottom number is up; negative is down.

Note: When you see capital letters on coordinate axes, that represents a specific point, like representing the origin. When you see that represents the length between the two points and If there is an arrow on top like that represents the vector that takes you from the point to the point

Notice how on the above grid an arrow is used also when representing a vector, which is why direction is also described. Reversing a vector keeps the same magnitude but points it in the opposite direction. In symbols, This little fact does a lot of heavy lifting in the harder questions later.

Before we go further, let's get the notation straight, because exam questions mix all of it. As we have already discussed, points are labelled with capital letters while a vector itself is usually given a lower-case letter or the point direction notation, e.g. used previously.

There are three common ways to write the same vector, and you should be comfortable with reading all three:

Three Notations for One Vector

With an arrow over two capitals: which means “the vector from to ”

As a bold lowercase letter: which is common in printed textbooks and exam papers.

As an underlined lower-case letter: which is the version you write by hand, since you can't easily make a letter bold with a pen. Watch any video below and you will see NeilDoesMaths write with underlined letter notation.

Throughout these notes we'll use for a vector between named points, and bold for vectors given their own name.

Here's the single most useful idea in the whole topic. If you want to get from to you don't have to go directly, you can take a detour through another point as shown in the diagram below. Travelling from to then from to lands you in exactly the same place as going directly from to

In the triangle above, we can see that

and

Adding these vectors gives

This matches the direct movement from to which, on the graph, is square to the left and then squares up.

Therefore,

The Path Rule (Triangle Law)

To add vectors, join them nose-to-tail. Reading across the outer letters give the start and the finish and the middle letter must match up.

Now, we put the path rule to work inside a shape. The trick is always the same: find a route made only of vectors you already know and then add them up. A regular hexagon is a favourite exam shape because its opposite sides are equal and parallel, so they represent the same vector, but they’re just pointing the opposite way when you travel around the shape.

The typical vector question on a GCSE paper usually comes in two parts. Part a) asks for a straightforward vector, often a side of a triangle. Then, part b) introduces a point that divides a line in a given ratio, and you build a route to it using your answer from part a). Get the ratio step right and these become very doable.

The key skill is turning a ratio into a fraction. If a point lies on so that then is out of every parts of the way along.

Dividing a Line in a Given Ratio

If then:

The denominator is the total number of parts; the numerator is the part you've reached.