Introduction to Vectors (Cambridge (CIE) O Level Additional Maths): Revision Note

Basic Vectors

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

Magnitude of a Vector

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

Position Vectors

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