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

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  is written  when typed (or  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  or   then  or 

• 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