Complex Analysis: Proving Vector z1 Parallel to z2

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Homework Help Overview

The problem involves demonstrating that the vector z1 is parallel to z2 if and only if the imaginary part of the product of z1 and the conjugate of z2 is zero. The context is within complex analysis, focusing on properties of complex numbers and their geometric interpretations.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss converting complex numbers to polar form and calculating the imaginary part of the product. There are attempts to relate the result to trigonometric identities, and questions arise about the implications of the derived equations for proving parallelism.

Discussion Status

Participants are actively engaging with the problem, recalculating values, and exploring trigonometric relationships. There is no explicit consensus yet, but some guidance is being offered regarding the interpretation of the results in terms of angle relationships.

Contextual Notes

There is a note about the potential for misunderstanding in the calculation of the imaginary part, and the discussion includes references to trigonometric formulas that may influence the reasoning process.

shannon
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Homework Statement


Show that the vector z1 is parallel to z2 if and only if Im(z1z2*)=0

note: z2* is the complement of z2


Homework Equations





The Attempt at a Solution


I would probably convert z to polar form.
so, z1=r1(cos Ѳ1+isin Ѳ1)
z2=r2(cos Ѳ2+isin Ѳ2)
so, z2*=r2(cos Ѳ2-isin Ѳ2)

Then, I would plug it into Im(z1z2*)=0

so, Im(r1(cos Ѳ1+isin Ѳ1)r2(cos Ѳ2-isin Ѳ2))

which is: r1r2sin Ѳ2sin Ѳ2

But I'm not sure where to go from here...

PLEASE HELP!


 
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Your final answer for the imaginary part isn't correct. If you do it right, it might just resemble the trig addition formula for sin(theta1-theta2).
 
Ok, so I recalculated my final value for the imaginary part and got...

0=r1r2(sin Ѳ1cos Ѳ2-cos Ѳ1sin Ѳ2)

So then I got:
sin Ѳ1cos Ѳ2=cos Ѳ1sin Ѳ2

So from here, do I just try to show that Ѳ12 to show that the vectors are parallel?
If so, how would I go about doing that?
 
Doesn't that look like a trig formula for the sine of the difference of two angles?
 

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