# Complex analysis

1. Jan 27, 2009

### shannon

1. The problem statement, all variables and given/known data
Show that the vector z1 is parallel to z2 if and only if Im(z1z2*)=0

note: z2* is the complement of z2

2. Relevant equations

3. 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!!

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jan 27, 2009

### Dick

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).

3. Jan 27, 2009

### shannon

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?

4. Jan 27, 2009

### Dick

Doesn't that look like a trig formula for the sine of the difference of two angles?

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