complex no qs (1 Viewer)

[skepticality]

1) Let u, v, w be three complex numbers such that |u| =|v|=|w|=1 and u + v + w =0. Show that the points representing u,v,w are vertices of an equilateral triangle inscribed in a unit circle.

2) Let w, z be two non-zero comples numbers. If wZ+Wz = 0, show that w/z is purely imaginary. What is the relationship between the vectors representing w and z?
(nb, where W is the conjugate of w and same for Z and z)

3) Let P(z₁), Q(z₂) and R(z₃) be three points in the Argand Plane. If a = z₂ - z₁, b = z₃ - z₁ and a, b are respectively represented by the points A,B, show that:

i) [Triangle] PQR and [Triangle] OAB are congruent.
ii) a² - ab + b² = 0 if and only if [triangle] OAB is equilateral.
iii) (z₁-z₂)^-1 + (z₂-z₃)^-1 - (z₃-z₁)^-1 = 0 if and only if [triangle] PQR is equilateral

thanks in advance.

 

[Riviet]

1) Since the moduli of each are 1, then they are all on the unit circle. If u+v+w=0 this means when you add the vector of each, they will cancel each other out and end up at the origin. Therefore the vectors must be equally spaced around the circle (120^o or 2pi/3 apart). They are actually the roots of a complex number such as 1 since they are all evenly spread around the unit circle. Therefore the distances from a point to another will be equal. Since they are evenly spread out, their angles formed with each other will be equal (60 degrees). Therefore, the points form an equaliteral triangle. You will probably need to show this by algebra, and explain that the points are the complex roots of a number with moduli 1.

2) Let w=x+iy and z=a+ib
then (x+iy)(a-ib)+(x-iy)(a+ib)=0
expanding and simplifying you obtain:
ax+by=0 *
w/z=(x+iy)/(a+ib)
=(ax+by-bxi+ayi)/(a²+b²)
=(ay-bx)i/(a²+b²) [from *]
which is purely imaginary. Not sure what the relationship between z and w is lol.

 

[brett86]