Complex numbers help. (1 Viewer)

[conics2008]

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In an Argand diagram, the point A represents the complex number z, the point C​

represents the complex number​

w and the point B represents the complex number z + w​

with​

O being the origin. Describe the geometric properties of the quadrilateral OABC,​

providing full reasoning for your answer, given that z − w = 2i(z + w).

Hi I just came across this question and went blank, I drew the thing but the last thing i got confused about it.

Can someone tell me how to approach questions like this. Thank you.

 

[Iruka]

Well, it is a parallelogram, of course.

One of the diagonals is z+w, the other is z-w translated a bit.

So, the fact that z-w = 2i(z+w) means that the two diagonals cross each other at right angles and one is twice the length of the other, as multiplication by a complex number corresponds to a rotation + a scaling. That tells you that the parallelogram is actually a rhombus. (And you should be able to figure out the size of the angles at the vertices by considering the lengths of the diagonals.)

 

[waxwing]

Can someone tell me how to approach questions like this. Thank you.

On the approach:
Use the known relationships for modulus and argument of product and quotient:


arg(z_1/z_2) = arg(z_1) - arg(z_2) (+2*k*pi, if you must...)
|z_1/z_2| = |z_1|/|z_2|

Thus in the case the difference in the arguments of z-w and z+w is equal to the argument of 2i, which is pi/2.
You get the picture

Another thing to bear in mind is to treat complex numbers as in some ways like vectors.
Therefore when you look at something like arg(z-w), you're looking at the angle from the horizontal projected rightwards from w, with the line from w to z. It's as if you're shifting the origin to w, and then calculating z's argument from there.
Btw, to go a little further on Iruka's points, I think you should get something like @ = cos^-1(1/3)

Hope it helps

 

[ronnknee]

Well, it is a parallelogram, of course.

One of the diagonals is z+w, the other is z-w translated a bit.

So, the fact that z-w = 2i(z+w) means that the two diagonals cross each other at right angles and one is twice the length of the other, as multiplication by a complex number corresponds to a rotation + a scaling. That tells you that the parallelogram is actually a rhombus. (And you should be able to figure out the size of the angles at the vertices by considering the lengths of the diagonals.)

A kite, not a rhombus

 

[Iruka]

I think you'd better go and revise your special quads.

Or actually read all that I have written in that post.

A rhombus is both a parallelogram and a kite.