Complex numbers 3

1
Complex numbers (3)

• Loci on Argand Diagrams

Starter
Sketch on an Argand diagram the points that
satisfy
2
Difference of two complex numbers
What does z2 z1 represent?
Im
This represents the distance between two complex
numbers z1 z2.
z2
z2 - z1
What is the value of z2-z1 in the diagram
Can I change the value of z2 without changing
z2-z1. Draw these points onto an Argand diagram
z1
What shape does it make?
Re
What is the centre of the shape?
What do you think arg(z2 z1) represents?
3
Loci

• Draw an argand diagram showing the set of points
  for which z 3 4j 5
• Solution
• First re-arrange the question
• z (3 4j) 5
• From the previous slide this represents a
  constant distance of 5 between the point (3,4)
  and z.
• This will give a circle centre (3,4)

4
Loci

• Draw Argand diagrams showing the sets of points z
  for which
• i) arg z p/3
• ii) arg (z - 2) p/3
• iii) 0 arg (z 2) p/3

5
Loci

• Sketch the locus of points z 2 z i.
• Describe in words what this means.
• z 2 is the distance from the co-ordinate
  A(-2,0) to any point z x iy. z 2 z
  (-2)
• z i is the distance from the co-ordinate
  B(0,1) to any point z x iy.
• As the position of z is the same then this means
  the distance from z to (-2,0) must be the same
  as the distance from z to (0,1).

6
Loci

• In other words z is equidistant from bath
  co-ordinates.
• This is just like a perpendicular bisector from
  GCSE.

Im
(0,1)
(-2,0)
Re
7
Loci - Example

• With the aid of a sketch, explain why there is
  no complex number which satisfies both of these