1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 41

6. Show that if the four points z 1 , z 2 , z 3 , z 4 are the vertices of a convex
   quadrilateral, then

where equality holds iff the quadrilateral is inscribed in a circle

(Ptolemy's theorem).

7. Show that the area of a triangle with vertices Z1, z2, za is given by

1 1 1

A=

i

• ,.?1 Z2 Z3
  4
  z1 z2 za

8. Let Z1 = a + bi and z 2 = c + di, and suppose that Arg z 1 f:. Arg z 2.
   Let OM be the vector corresponding to the product cz 1 and ON be
   the vector corresponding to az 2 • Construct the perpendicular to OM

at the point Mand the perpendicular to ON at the point N, and let

P be the point of intersection of those perpendiculars. Prove that the

vector OP represents the product z 1 z 2 •

9. The dot product and the cross product of two nonzero complex numbers
   z 1 and z 2 are defined as follows:
   Dot product: z1 • z2 = lz1llz2I cosB = x1x2 + Y1Y2

= Re(z1z2)

Cross product: Z1 x Z2 = lz1 llz2 I sine= X1Y2 - X2Y1

= Im(z1z2)

where 8 is the measure of the angle from the vector OP representing

z 1 to the vector OQ representing z 2 • Prove the following.

(a) z1z2 = (z1 • z2) + i(z1 x z2) = lz1llz2leill

(b) OP is perpendicular to OQ iff Z1 • Z2 = o, or z1z2 + Z1Z2 = 0.

( c) The line supporting OP coincides with the line supporting OQ iff

Z1 x Z2 = o, or Z1Z2 - Z1Z2 = 0.

(d) The length of the projection of OP on OQ is lz1 · z2l/lz2I·

(e) The area of the parallelogram with sides OP and OQ is lz1 X z2I·

10. Let ). be a given imaginary number.
    (a) Prove that all triangles with the vertices z 1 , z 2 , z 3 and such that
    (za - z1)/(z2 - z1) = >. are similar.
    (b) For what values of >. do we get equilateral triangles?
    (c) What happens if>. is chosen to be real?

( d) Assuming that z 2 and z 3 are fixed, find the locus of z1 when >. = it

(t real).