How do the lengths and angles of vectors relate in complex multiplication?

  • #1
bfusco
128
1

Homework Statement


Consider a vector z defined by the equation z=z1z2, where z1=a+ib, z2=c+id.
(a) show that the length of z is the product of the lengths of z1 and z2.
(b) show that the angle between z and the x-axis is the sum of the angles made by z1 and z2 separately.

The Attempt at a Solution


(a) i want to just do regular multiplication. (a+ib)(c+id)= ac-bd+i(ad+bc) however i don't see how that would show the length of z is the product of z1 and z2, all i did was multiply.

my next idea would be to take the magnitudes of z1 and z2 and multiply them. so, (√[a^2+(ib)^2]) * (√[c^2+(id)^2]) = (√[a^2-b^2])(√[c^2-d^2]).

(b)this would depend in part on which attempt of part (a) is correct. this is because depending on the correct way vector z is represented with its components the angle is going to be different.

Thank you in advance.
 
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  • #2
I believe for part (a) you are trying to do

so you can verify that based on what you have you know that if then

(b) Use the fact that I think.
 
  • #3
bfusco said:

Homework Statement


Consider a vector z defined by the equation z=z1z2, where z1=a+ib, z2=c+id.
(a) show that the length of z is the product of the lengths of z1 and z2.
(b) show that the angle between z and the x-axis is the sum of the angles made by z1 and z2 separately.

The Attempt at a Solution


(a) i want to just do regular multiplication. (a+ib)(c+id)= ac-bd+i(ad+bc) however i don't see how that would show the length of z is the product of z1 and z2, all i did was multiply.
Let and = angle between and the x-axis. Using basic trig, you should be able to see that you can write
Try plugging that into your expression for the product.

my next idea would be to take the magnitudes of z1 and z2 and multiply them. so, (√[a^2+(ib)^2]) * (√[c^2+(id)^2]) = (√[a^2-b^2])(√[c^2-d^2]).
You don't want to include the factor of when squaring. The magnitude of a complex number is given by where is the conjugate. If you work that out, you'll see the drops out.

(b)this would depend in part on which attempt of part (a) is correct. this is because depending on the correct way vector z is represented with its components the angle is going to be different.

Thank you in advance.
 

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