Magnitude of a Vector (Edexcel IGCSE Further Pure Maths): Revision Note

Magnitude of a Vector

How do I find the magnitude of a vector?

• The magnitude of a vector is its length (distance)

  • It is also called the modulus

• The magnitude of is written

  • The magnitude of a is written |a|

• In component form, the magnitude is the hypotenuse of a right-angled triangle

  • Use Pythagoras' theorem to find it

  • The magnitude of is

    • 

    • where 

How do I find harder magnitudes?

• The magnitude of a sum of vectors is not equal to the sum of the magnitudes

  • 
    

    • Work out a + b (or a - b) first, then find its magnitude 

• You may need to form an equation

  • For example, |a| = 5 where a = 4i + xj

    • 

    • This solves to give 

How do I find the magnitude of a displacement vector?

• You can use coordinate geometry to find magnitudes of displacement vectors from A  to B

  • From the position vectors of A  and B  you know their coordinates

    • If  ,  then point A has coordinates

    • If  ,  then point B has coordinates 

  • The distance between two points is given by 

    • So 

  • For example, if points A and B have position vectors  and  respectively

    • then  

• Alternatively, you could find  by

  • first using  to find in vector form

    • and then calculating its magnitude directly

  • See the Worked Example below 

Unit Vectors

What is a unit vector?

• A unit vector is a vector of length 1

• Any vector, a, can be made a unit vector by dividing it by its magnitude,

  • This will result in a vector of length 1

    • It still points in the same direction as a

• For example, to make 3i – 4j a unit vector

  • find its magnitude,

  • divide by its magnitude

    • 

    • Each component is divided by the magnitude