I was looking back at some complex numbers stuff and i can't figure out how this proves it:
Anyway the Question is: "Use the vector representation of z1 and z2 on an Argand diagram to show that:
If |Z1| =|Z2| then (Z1+Z2)/(Z1-Z2) is imaginary."
Well so far i got that it's a rhombus as all sides are equal so therefore z1+z2 and z1-z2 intercept at right angles. I'm not sure how it proves it's imaginary.
Does anyone know how this proves it or is there another step i need to do?
Also Z1 and Z2 aren't given.
(|Z1| means the modulus - or length of Z1, i'm not sure if notation is the same around the world)
Well i think i've kind of figured it out to anyone that actually gets the OP. Since they intercept at right angles there is a rotation of 90 degrees from one to the other, and since iz is z rotated 90 degrees i'm pretty sure it's got something to do with that.
[editline]05:50AM[/editline]
By the way this is only Question 2 part A :v:
42. /end thread.
Seriously though, I have no clue.
If I knew what an Argand diagram was, I'd be able to help.
[URL]http://www.tutorvista.com/content/math/number-theory/complex-numbers/complex-number-geometrical.php[/URL]
And thanks for the replies anyway <3
[editline]06:15AM[/editline]
The pictures are of Argand diagrams, the 2nd one is a better example as it is related to the question. The X axis is for real numbers the Y axis is for imaginary ones
[QUOTE=GunsNRoses;19696994]I was looking back at some complex numbers stuff and i can't figure out how this proves it:
Anyway the Question is: "Use the vector representation of z1 and z2 on an Argand diagram to show that:
If |Z1| =|Z2| then (Z1+Z2)/(Z1-Z2) is imaginary."
Well so far i got that it's a rhombus as all sides are equal so therefore z1+z2 and z1-z2 intercept at right angles. I'm not sure how it proves it's imaginary.
Does anyone know how this proves it or is there another step i need to do?
Also Z1 and Z2 aren't given.
(|Z1| means the modulus - or length of Z1, i'm not sure if notation is the same around the world)[/QUOTE]
If (z1) = (z2) it's impossible, I'm doing precisely these things in maths right now.
It's not saying z1 = z2 it's saying the modulus of them are equal
[editline]06:44AM[/editline]
Regardless, props to being the first person to actually get what's going on.
Aha, I remember this stuff... From like, 2 years ago.
I'm reasonably sure this would work as a proof seeing as the question asked you to show using the argand diagram. Maybe. hahaha
From here on I'll call:
z1 + z2 = A
z1 - z2 = B
You've done most of the work by showing that A and B intersect at right angles! As long as you've done this mathematically, by assigning arbitrary vectors for z1 and z2, such as z1 = a + bi and z2 = x + yi, then using the dot product to get another expression. You'll be able to show that this expression is zero, using the fact that
a^2 + b^2 = x^2 + y^2
I'm guessing you've already done this though, because you mentioned that they're at right angles :)
Mark A and B on your argand diagram, and label the argument of each (theta 1, theta 2 - or whatever)
you know that the angle between them is 90 degrees.
Now you have to know that dividing one complex number by another (or multiplying) always has the same geometric effect. In this case we're dividing, and the resulting complex number will have a magnitude equal to the magnitude of the numerator divided by the denominator, and an argument equal to the argument of the numerator subtract that of the denominator.
You can show on your diagram that this will always result in an argument of 90 or 270 degrees (in fact, it might be 90 every time, I kind of rushed through this), right on the imaginary axis. So for whatever z1 and z2 you begin with, the result of
A / B will never have a real component.
I did this rather roughly, so please point out any likely screw ups or anything that needs clarifying :)
I started off trying to do this mathematically and not on the argand diagram, if you do as well you'll see that you end up with a complex number that has no real part.
The answer is 47x2
^This man speaks the truth.
The answer is that there's an elephant in the way.
[QUOTE=Little Green;19700055]You can show on your diagram that this will always result in an argument of 90 or 270 degrees (in fact, it might be 90 every time, I kind of rushed through this), right on the imaginary axis. So for whatever z1 and z2 you begin with, the result of
A / B will never have a real component.[/QUOTE]
Thanks mate, and yeah i got both your PM's. This bit is the hard bit, i could do it mathematically but i don't know if it's possible using geometric proofs to show that it'l always end up on the Y axis. Anyway cheers
[QUOTE=GunsNRoses;19696994]"Use the vector representation of z1 and z2 on an Argand diagram to show that:
If |Z1| =|Z2| then (Z1+Z2)/(Z1-Z2) is [b]imaginary[/b]."[/QUOTE]
Obviously,
[img]http://img200.imageshack.us/img200/1892/zisimaginary.png[/img]
[QUOTE=GunsNRoses;19701957]Thanks mate, and yeah i got both your PM's. This bit is the hard bit, i could do it mathematically but i don't know if it's possible using geometric proofs to show that it'l always end up on the Y axis. Anyway cheers[/QUOTE]
By the dot product, the angle between z1 + z2 and z1 - z2 is 90 degrees. So it follows that the argument of (z1 + z2)/(z1 - z2) is also going to be 90, because the result of dividing two complex numbers is another with an argument equal to the difference between them.
If the argument is 90, it has to lie somewhere on the imaginary axis.
You may just have to show that this property of division is true, but if you've covered polar form then I don't think you really bother showing every time.
[QUOTE=Little Green;19702321]By the dot product, the angle between z1 + z2 and z1 - z2 is 90 degrees. So it follows that the argument of (z1 + z2)/(z1 - z2) is also going to be 90, because the result of dividing two complex numbers is another with an argument equal to the difference between them.
If the argument is 90, it has to lie somewhere on the imaginary axis.
You may just have to show that this property of division is true, but if you've covered polar form then I don't think you really bother showing every time.[/QUOTE]
It just clicked then, thanks.
Surely if Z1 = Z2, Z1 - Z2 = 0. That means that in the equation you divide Z1 + Z2 by 0.
Where JohnnyMo at.
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