Vectors (AQA GCSE Maths): Revision Notes

Vectors

What are vectors?

Vectors describe how to move from one point to another, showing both the distance you travel and the direction you go. At first glance, they might seem a bit unusual, but once you understand the key concepts, they become much more manageable.

Vector notation

There are several different ways we can write vectors, and it's important to know them all for your exams.

Column vectors are written as two numbers in brackets, one above the other. The top number shows horizontal movement (positive means right, negative means left), and the bottom number shows vertical movement (positive means up, negative means down). For example, the vector $\begin{pmatrix} 2 \\ -5 \end{pmatrix}$ means move 2 units right and 5 units down.

Letter notation is used in exam questions where vectors are represented by letters like a or b. In handwritten work, you should always underline these letters to show they represent vectors.

Position vectors like $\overrightarrow{AB}$ show movement from point A to point B, with the arrow indicating the direction.

Multiplying vectors by scalars

When we multiply a vector by a number (called a scalar), we change the vector's size but keep it pointing in the same direction - unless the number is negative, which flips the direction completely.

Multiplying by a positive number scales the vector up or down while maintaining its direction. For instance, if vector a represents a certain movement, then $2\mathbf{a}$ represents the same movement but twice as long.

When we multiply by a negative number, the vector not only changes size but also reverses direction. So $-1.5\mathbf{a}$ would be 1.5 times longer than a but pointing in the opposite direction.

Adding and subtracting vectors

Vector addition and subtraction help us describe movements between different points by combining known vectors.

Vector addition $(a + b)$ means "go along vector a, then go along vector b". This creates a path from your starting point to your final destination.

Vector subtraction $(c - d)$ means "go along vector c, then go backwards along vector d". The minus sign tells us to travel in the opposite direction to vector d.

When working with column vectors, we add or subtract them by combining the top numbers together and the bottom numbers together.

Proving points lie on straight lines

One of the most important applications of vectors is proving that three points are collinear (lie on the same straight line).

The key principle is that if three points X, Y, and Z lie on a straight line, then the vector $\overrightarrow{XY}$ must be a scalar multiple of vector $\overrightarrow{YZ}$. This is because both vectors point along the same line, just with different lengths.

Ratios in vector problems

Ratios help us find the lengths of different sections along a straight line when we know how the line is divided.

If we know that $XY : YZ = m : n$, this tells us that $\overrightarrow{XY} = \frac{m}{m+n}\overrightarrow{XZ}$ and $\overrightarrow{YZ} = \frac{n}{m+n}\overrightarrow{XZ}$. This is particularly useful when dealing with parallelograms and other geometric shapes where we need to find unknown vectors.

When working with ratios, we can use information about parallel sides and the given ratio to find missing vectors step by step.