Problem Solving using Vectors (Cambridge O Level Additional Maths)

What problems may I be asked to solve involving vectors?

• Showing that two lines or vectors are parallel
  
  • Two vectors are parallel if they are scalar multiples of each other
  • i.e.   where  is a constant
    • See Vector Addition
• Finding the midpoint of two (position) vectors
• Showing that three points are collinear
  
  • Collinear describes points that lie on the same straight line
    • e.g.  The points  and  all lie on the line with equation 
    • Vectors can be used to show this, and similar, results
• Results concerned with geometric shapes
  • Shapes with parallel lines are often involved
    • e.g.  parallelogram, rhombus
  • These often include lines or vectors being split into ratios
    • e.g.  The point  lies on the line  such that

How do I find the midpoint of two vectors?

• If the point  has position vector and the point  has position vector
  • the position vector of the midpoint of  is 
• This can be derived by considering
  • 
    • using the result from Vector Addition 
  • If  is the midpoint of then
    • 
  • Therefore, the position vector of the midpoint,  is  
    • 
       

How do I show three points are collinear?

• Three points are collinear if they all lie on the same straight line
• There are two ways to show this for three points, and say
  • Method 1
    Show that  where  is a constant
    i.e.  show that  and  are scalar multiples of each other
    • As the vectors are scalar multiples they will have the same direction (and so be parallel)
    • So as both vectors start at point , they must be collinear
  • Method 2
    Show that   AND  that point lies on both the vectors  and 
• Which method you should use will depend on the information given and how you happen to see the question

How do I solve problems involving geometric shapes?

• Problems involving geometric shapes involve finding paths around the shape using known vectors
  • there will be many other vectors in the shape that are equal and/or parallel to the known vectors
• The following grid is made up entirely of parallelograms, with the vectors  and  defined as marked in the diagram:

• Note the difference between "specific" and "general" vectors
  • The vector in the diagram is specific and refers only to the vector starting at and ending at
    • However, the vector  is a general vector
      • any vector the same length as  and parallel to it is equal to
      • e.g. 
    • Vector is also a general vector
      • e.g.   
  • There will also be vectors in the diagram that are the same magnitude but have the opposite direction to or 
    • e.g.  
• There are also many instances of the vector addition result 
  • e.g.  
• There are many scalar multiples of the vectors or 
  • e.g.  
• Using a combination of these it is possible to describe a vector between any two points in terms of  and