Introduction to Vectors (Cambridge O Level Additional Maths)

What is a vector?

• Vectors represent a movement of a certain magnitude (size) in a given direction
  • For example: two objects with velocities of 7 m/s and ‑7 m/s are travelling at the same speed but in opposite directions
• You should have already come across vectors when translating functions of graphs
• They appear in many contexts of maths including mechanics for modelling forces
• A vector in two directions has components in the direction of the x- and y- axes
  • Vector quantities can have positive or negative components
• Vectors can be represented in different ways such as a column vector or as an i and j unit vector
• Some examples of vector quantities you may come across are displacement, velocity or acceleration

How do you find the magnitude of a vector?

• The magnitude of a vector tells us its size or length
• The magnitude of the vector is denoted 
  • The magnitude of the vector a is denoted |a|
• The magnitude of a vector can be found using  Pythagoras’ theorem
• The magnitude of a vector  is found using
  • 
  • where 

What is a unit vector?

• A unit vector has a magnitude of 1
• The vectors and  are unit vectors
  • the direction of  is in the positive -direction
  • the direction of  is in the positive -direction
• To find a unit vector in the direction of a given vector divide the vector by its magnitude
  

What is a position vector?

• Position vectors describe the position of a point in relation to the origin
• They are different to displacement vectors which describe the direction and distance between any two points
• The position vector of point A is written with the notation a =  
  • The origin is always denoted O
• The individual components of a position vector are the coordinates of its end point
  • For example the point with coordinates (3, -2) has position vector 3i – 2j

How do I find the distance between two points using vectors?

• The distance between two points is the magnitude of the vector between them

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