Complex Analysis: Proving Vector z1 Parallel to z2

In summary, the problem is trying to show that two vectors, z1 and z2, are parallel if and only if the imaginary part of their product, z1z2*, is equal to 0. To solve this, the conversation suggests converting the vectors to polar form and plugging them into the equation. After some calculations, the final answer for the imaginary part is incorrect and needs to be redone using the trigonometric addition formula. From there, the conversation suggests trying to show that the angles of the two vectors are equal in order to prove that they are parallel.
  • #1
shannon
11
0

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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  • #2
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
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
Doesn't that look like a trig formula for the sine of the difference of two angles?
 

Related to Complex Analysis: Proving Vector z1 Parallel to z2

1. What is complex analysis?

Complex analysis is a branch of mathematics that deals with the study of complex numbers and the functions that are defined on them. It involves the application of calculus and other mathematical tools to understand the properties and behavior of complex functions.

2. How do you prove that two vectors are parallel in complex analysis?

In order to prove that two vectors, z1 and z2, are parallel in complex analysis, you can use the definition of parallel vectors which states that two vectors are parallel if their directions are the same. This can be shown by proving that the ratio of their corresponding components is equal.

3. What is the importance of proving vector z1 parallel to z2 in complex analysis?

Proving that two vectors are parallel is important in complex analysis because it helps us understand the geometric properties of complex functions. It also allows us to simplify complex problems by reducing them to more manageable and easier to understand forms.

4. What are the different methods used to prove parallelism in complex analysis?

There are several methods that can be used to prove parallelism in complex analysis. These include using the definition of parallel vectors, using the properties of complex numbers, and using geometric properties such as the dot product or cross product.

5. Can you provide an example of a proof of parallelism in complex analysis?

Yes, for example, to prove that vector z1 = 2+3i is parallel to vector z2 = 4+6i, we can use the definition of parallel vectors and show that their corresponding components have the same ratio. In this case, both z1 and z2 have a real component of 2 and an imaginary component of 3, which have a ratio of 2:3. Similarly, both z1 and z2 have a real component of 4 and an imaginary component of 6, which also have a ratio of 2:3. Therefore, we can conclude that z1 and z2 are parallel in complex analysis.

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