Geometric Complex Numbers (1 Viewer)

[NewiJapper]

Let z = 3+ 2i and w =1+ 3i . Using complex numbers, find all complex numbers v such
that z, w and v form the vertices of an isosceles triangle in which the length of side zw​

equals the length of side zv and the base angles are 67.5 degrees .

I've attempted this a couple of ways but I can't seem to get the write answer. Any help?​

 

[khfreakau]

Okay, first of all, you know the locus of v is (x-3)^2 + (y-2)^2 = 5, so any point v has to satisfy those conditions. (not really necessary, but nice to get an idea of what's going on)

67 degrees? Really? That's slightly troll, so i'll use 67.5 so that i can get an exact value. You can just sub in 67 later if need be.
First you can find the vector wz, where w is the head and z is the tail. Then, with that vector, wouldn't you just multiply it by cis67 (3pi/8 for simplicity purposes) and -cis3pi/8?

 

[funnytomato]

isn't ∠WZV 180-67-67=46 ?

 

[funnytomato]

then after multiplying zw by cis( whatever that angle is) , you get zv, then ov = oz+zv

 

[kooliskool]

First you can find the vector wz, where w is the head and z is the tail. Then, with that vector, wouldn't you just multiply it by cis67 (3pi/8 for simplicity purposes) and -cis3pi/8?

Erm, the second complex number should be multiply by cis(-46) (in degree), not -cis46 though, it makes a lot of difference......

 

[funnytomato]

i got v = (3-2cos46°-sin46°) + i(2+cos46°-2sin46°) or v = (3-2cos46°+sin46°) + i(2+cos46°+2sin46°)
any one wanna compare answers?

 

[kooliskool]

Yep, I got the same as u.

 

[khfreakau]

Erm, the second complex number should be multiply by cis(-46) (in degree), not -cis46 though, it makes a lot of difference......

My bad, it's been a while since I've done complex numbers

 

[hscishard]

Okay, first of all, you know the locus of v is (x-3)^2 + (y-2)^2 = 5

I get your idea but it's not expressed correctly. If the locus was that circle, then that would be your answer to that question. You should've said that the possible numbers for v lies on (x-3)^2 + (y-2)^2 = 5
sorry if I'm not making sense

 

[funnytomato]

I get your idea but it's not expressed correctly. If the locus was that circle, then that would be your answer to that question. You should've said that the possible numbers for v lies on (x-3)^2 + (y-2)^2 = 5sorry if I'm not making sense

yes,you're making sense

 

[NewiJapper]

Yes it was 67.5 degrees sorry, my bad!

Thanks for the help though guys! I initially wasn't thinking with vectors to solve this problem and ended up with some retarded ellipse haha

Thanks!

 

[kfnmpah]

i got v = (3-2cos46°-sin46°) + i(2+cos46°-2sin46°) or v = (3-2cos46°+sin46°) + i(2+cos46°+2sin46°)
any one wanna compare answers?

I'm doing this question now, too. why, in the answer, is it 46°, not 67°?

 

[kooliskool]

Because If you draw the diagram properly, the angle between the vector zw (z is tail, w is head) and vector zv should be 45 (since they said the original quesiton is 67.5), because it should be the angle between the two equal sides.

Therefore you are rotating the vector by 45, not 67.5.

Hope that helps.