complex# prob (1 Viewer)

[Faera]

Yup, I'm stuck, yet again:

(i) State a condition for z, w, v (all complex) to form the vertices of an equilateral triangle on the Argand Diagram.

(ii) If z = 1 + 3i, find a complex number w, so that z, w and the origin O form the vertices of an equilateral triangle.

Thanks for taking a look, and any help would be good


p.s. Cake for whoever explains it to me first!!

 

[Grey Council]

first one:
look below at Affinities reply OR:
(z-w)/(w-v) = (w-v)/(v-z) = (v-z)/(z-w) = cis(+-pi/3)

second one:
cos(arctan3+pi/3) + isin(arctan3+pi/3)
expand that all out, you'll get exact form.
i hate trig. i'm not even gonna try. lol

I give up. no cake for me, i s'pose. berloody, cm_tutor will get it again. meh

 

[Affinity]

(i) notice how it says 'a condition'? so a perfectly legitimate answer would be z,w,v are distinct cube roots of unit.

the general condition would be (z-w),(w-v),(v-z) are k, xk,x^2k in any order,where x is a complex cube root of unity

ii) rotate 1+3i clockwise or anticlockwise 60 degrees about the origin.. which is same as multiplying by cis(Pi/6) and cis(-Pi/6)

will leave you to do it

 

[CM_Tutor]

Another general condition would be |z - w| = |w - v| = |v - w|

Affinity, I think you mean pi / 3, not pi / 6.

 

[Faera]

Hmm...
Grey_C, I don't really get what you're saying... perhaps i'll get you to explain it to me over msn later on... -.-'

Affinity, what you're saying's true, but then what if the equilateral trianlge formed isnt actually formed with the origin as center? Because for part (ii), one of the vertices is actually O...

CM- by |z - w| = |w - v| = |v - w|, do you mean |z - w| = |w - v| = |v - z| ?

 

[CM_Tutor]

Originally posted by Faera
CM- by |z - w| = |w - v| = |v - w|, do you mean |z - w| = |w - v| = |v - z| ?

Yes, you're right, I've made a typo on the last term. Oops.

As for the other part, there are two possible points, which I get as (1 - 3 * sqrt(3)) / 2 + (3 + sqrt(3))i / 2 or
(1 + 3 * sqrt(3)) / 2 + (3 - sqrt(3))i / 2

 

[Grey Council]

Originally posted by Faera
Hmm...
Grey_C, I don't really get what you're saying... perhaps i'll get you to explain it to me over msn later on... -.-'

Affinity, what you're saying's true, but then what if the equilateral trianlge formed isnt actually formed with the origin as center? Because for part (ii), one of the vertices is actually O...

CM- by |z - w| = |w - v| = |v - w|, do you mean |z - w| = |w - v| = |v - z| ?

hrm, your rotating the LINE 1+3i by 60 degrees. ^_^ your sliding the line. rotating about the origin. hope you know what i mean.