Vectors (AQA GCSE Maths): Revision Notes

Vectors

What are vectors?

A vector is a mathematical quantity that has both magnitude (size) and direction. Think of it as an arrow pointing from one place to another - the length of the arrow shows the magnitude, and which way it points shows the direction.

Vector notation

Vectors can be written in several different ways:

• Using a single letter: a
• Using two points: AB (from point A to point B)
• As a column vector: $\begin{pmatrix} 2 \\ 5 \end{pmatrix}$

Scalar multiplication

When you multiply a vector by a number (called a scalar), you change its length but keep the same direction.

This is useful when you need to make vectors bigger or smaller whilst maintaining their direction.

Negative vectors

If b is a vector, then -b is a vector with the same magnitude but opposite direction.

Adding vectors

You can add vectors using the triangle law. This means you trace a path along the vectors to find the new combined vector.

The steps are:

1. Draw the first vector
2. From the end of the first vector, draw the second vector
3. The resultant vector goes from the start of the first vector to the end of the second vector

Working with vectors in geometry

Vectors are particularly useful when working with geometric shapes like parallelograms.

In geometric problems:

• Trace the path from the start point to the end point
• If you go in the opposite direction to a vector, you subtract it
• Parallel vectors are equal (like opposite sides of a parallelogram)
• Always simplify your vector expressions as much as possible

Key techniques for geometric vector problems

Column vector operations

When working with column vectors:

Multiplication by a scalar: $2 \times \begin{pmatrix} 2 \\ a \end{pmatrix} = \begin{pmatrix} 4 \\ 2a \end{pmatrix}$

Addition: $\begin{pmatrix} c \\ d \end{pmatrix} + \begin{pmatrix} e \\ f \end{pmatrix} = \begin{pmatrix} c+e \\ d+f \end{pmatrix}$

Summary