Vectors and a Hexagon

This item is taken from IGCSE Mathematics (0580) Paper 22 of October/November 2013.


To deal with this item, let us summarize the details of the problem:
   (a) Syllabus area: Vectors
   (b) Specific topic: Vectors, Position Vectors, Addition/Subtraction of Vectors
   (c) Concept Needed:

Let us start with the relations of the sides of a regular hexagon. 

Before we answer (a), let us start with the relations of the vectors given in the figure.

We cannot get the direct value of BE using the given vectors. Instead, we can find the value of OB first, then use the relation of the vectors BE = 2BO.

Considering vectors OC and CB, the resultant is vector OB. This uses the concept of triangle law of vector addition, since vector OC and CB are in a nose to tail position. 

However, we are finding BO. Hence, we can reverse the directions and find the resultant BO in terms of a and c.

We can find BE by using vector BO.

For (b), let us start with the relations of the vectors needed. 

Using the triangle law of subtraction of vectors.

For (iii), we should take note that it is a position vector. It means that the vector starts with O then ends with E. Hence, we are finding vector OE.

Using the relation of vector OE and vector BE,

Therefore,

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