Magnitude & Direction (Edexcel IAL Maths)

Spec point — 7.2

Magnitude & direction

What is the magnitude of a vector?

• 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 $\vert a \vert$ when typed (or | bottom a | when handwritten)

• 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

What is the direction of a vector?

• Vectors have opposite direction if they are the same size but opposite signs

  • e.g. if $a$ or $\vec{BC} = ( \begin{gathered} - 3 \\ 8 \end{gathered} )$ then $-a$ or $\vec{CB} = ( \begin{gathered} 3 \\ - 8 \end{gathered} )$

• 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)

How do I write a vector in component form?

• 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

Exam Hint

• 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.

Worked Example