Vectors on Triangles (Part 2)

This item is taken from IGCSE Mathematics (0580) Paper 2 of October/November 2007.
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To deal with this item, let us summarize the details of the problem:
   (a) Syllabus area: Vectors
   (b) Specific topic: Addition/Subtraction of Vectors, Position Vectors, Naming a Vector in Terms                                 of other Two Vectors
   (c) Concepts Needed: Triangle Law for Addition/Subtraction of Vectors

To answer (ii), let us recall the concept of position vectors. Position vectors will always start with a reference point, usually point O (or origin), and end at a specific point other than O. 

In this item, we are looking for the position vector of E. We will then connect O to E to form vector OE. Therefore, position vector of point E is the vector OE. 

Let us now write vector OE in terms of a and b. Take note that OA = a and OB = b. Let us also take note that the direction of the vectors matters. 

The first way of writing it in terms of a and b is to start with vector a. From O, go up to A until it reaches a point that it is horizontally aligned with E. It must be horizontally aligned so that the connecting vector will be parallel to vector b (vector OB). After that, go horizontal and parallel to vector b until you reach point E. This is shown in the figure below.

Now, let us determine the length of the connected vectors from O to E. The first vector if halfway (3 units out of 6) from O to A or 1/2 OA or 1/2 a. The second one is two units away from where the first vector ends. That is 2 out of 6 units of vector OB or 1/3 OB or 1/3 b.

Hence, the position vector of E is vector OE = 1/2 a + 1/3 b.   

The second way is to start with vector b. From O, go to B until it reaches a point that is diagonally aligned with point E. It must be diagonally aligned so that the connecting vector later will be parallel to vector a (vector OA). This is shown in the figure below.   

The first vector is 2 units out of 6 units of vector OB. That means 1/3 OB or 1/3 b. The connecting vector is 3 units out of 6 units of vector OA. That is 1/2 OA or 1/2 a. 

Hence, the position vector of E is vector OE = 1/3 b + 1/2 a or 1/2 a + 1/3 b. 

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