Working with Vectors (Cambridge O Level Maths)

Finding paths in vector diagrams

• It is important to be able to describe vectors by following paths through a geometric diagram
• 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 pointing in the same direction is equal to 
  • Similarly, any vector the same length as and pointing in the same direction is equal to
• Following a vector in the "wrong" direction (i.e. from head to tail instead of from tail to head) makes a general vector negative
  • So in the diagram above , but 
  • Similarly  and 
• Note in particular the vector :

• Getting from point to point  we have to go the 'wrong way' down  and then the 'right way' along

• It follows that:
• • 
• and of course then
  • 

• Keeping those things in mind, it is possible to describe any vector that goes from one point to another in the above diagram in terms of a and b

What are vector proofs?

• In vector proofs we use vectors, along with a few key ideas, to prove that things are true in geometrical diagrams
• Problem solving with vectors involves using these vector proofs to help us to find out additional information

Parallel vectors

• Two vectors are parallel if and only if one is a multiple of the other
• This tends to appear in vector proofs in the following ways:
  • If you find in your workings that one vector is a multiple of the other, then you know that the two vectors are parallel – you can then use that fact in the rest of the proof
  • If you need to show that two vectors are parallel, then all you need to do is show that one of the vectors multiplied by some number is equal to the other one
  • E.g. If and , then , therefore and  are parallel

Points on a straight line

• Often you are asked to show in a vector proof that three points lie on a straight line (ie that they are collinear)
• To show that three points ,  and  lie on a straight line,
  • show that the vectors connecting the three points are parallel,
    • for example, show that is a multiple of (and therefore parallel to) , or that  is a multiple of (and therefore parallel) to
  • as those two vectors are parallel and they share a common point it means that the three lines form a straight line

Vectors divided in ratios

• Be careful turning ratios into fractions in vector proofs!
• If a point  divides a line segment  in the ratio p : q, then: 

and

• • eg. In the following diagram, the point  divides  in the ratio 3: 5:

Therefore

and