Problem Solving using Vectors (Cambridge (CIE) O Level Additional Maths): Revision Note

Problem-solving using vectors

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