Working with Vectors | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Finding Vector Paths

Finding paths in vector diagrams

• Understanding vectors often involves tracing paths through geometric diagrams. 
• In the provided grid, comprised entirely of parallelograms, vectors a and b are identified as shown.

• Note the difference between "specific" and "general" vectors
  • The vector stack  in the diagram is specific and refers only to the vector starting at A and ending at B
  • However, the vector a is a general vector - any vector the same length as stack  and pointing in the same direction is equal to a
  • Similarly, any vector the same length as  and pointing in the same direction is equal to b
• 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 stack
  • Similarly
• 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

Vector Problem Solving

What are vector proofs?

• Vector proofs entail employing vectors, coupled with key concepts, to establish the validity of geometric assertions within diagrams.
• Problem-solving with vectors encompasses leveraging these vector proofs to deduce supplementary information and address challenges effectively.

Parallel vectors

• Two vectors are parallel if and only if one is a scalar multiple of the other.
• This principle manifests in vector proofs in the following ways:
  • If during your calculations you determine that one vector is a scalar multiple of the other, then you can conclude that the two vectors are parallel. This established fact can then be utilized in subsequent parts of the proof.
  • To demonstrate that two vectors are parallel, it suffices to show that one of the vectors, when multiplied by a scalar, equals the other vector.
• For instance: 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 A, B and C lie on a straight line,
  • show that the vectors connecting the three points are parallel,
    • for example, show that stack  is a multiple of (and therefore parallel to) stack  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: 
  • eg. In the following diagram, the point  divides  in the ratio 3 : 5:

Therefore