Magnitude & Direction

The magnitude of a vector is simply its size
It also tells us the distance between two points
You can find the magnitude of a vector using Pythagoras’ theorem
The magnitude of a vector $a$ is written $| a |$ when typed (or $| a |$ when handwritten)

Magnitude Direction Diagram 1a, AS & A Level Maths revision notes

To work out the unit vector in the direction of a given vector
A unit vector has a magnitude of 1
So to find the unit vector of a given vector, divide by its magnitude

Magnitude Direction Diagram 1b, AS & A Level Maths revision notes

Vectors have opposite direction if they are the same size but opposite signs
- e.g. if $a$ or $\vec{B C} = (\begin{matrix} - 3 \\ 8 \end{matrix})$ then $- a$ or $\vec{C B} = (\begin{matrix} 3 \\ - 8 \end{matrix})$

The direction of a vector is what makes it more than just a scalar
- Eg. two objects with velocities of 7 m/s and ‑7 m/s are travelling at the same speed but in opposite directions
Two vectors are parallel if and only if one is a scalar multiple of the other
For real-life contexts such as mechanics, direction can be calculated from a given vector using trigonometry (see Right-Angled Triangles)
It is usually calculated anticlockwise from the positive x-axis (unless otherwise stated eg. a bearing)

We have already seen that vectors can be written in different forms
Component form means writing a vector in terms of i and j components
Given the magnitude and direction of a vector you can work out its components and vice versa

Magnitude Direction Diagram 3a, AS & A Level Maths revision notes Magnitude Direction Diagram 3b, AS & A Level Maths revision notes

Diagrams can help, especially when working out direction – if there isn’t a diagram, draw one.
Remember, resolving a vector just means writing it in component form.

Magnitude Direction Example Solution, AS & A Level Maths revision notes

Magnitude & Direction (CIE A Level Maths: Pure 3)