Complex Analysis: Proving Vector z1 Parallel to z2

[shannon]

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!

 

[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).

 

[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 Ѳ₁=Ѳ₂ to show that the vectors are parallel?
If so, how would I go about doing that?

 

[Dick]

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