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Question from Vibrations and Waves by A.P. French Chapter 1

Consider a vector z defined by Z=Z_{1}Z_{2}, where Z_{1}=a+jb, Z_{2}=c+jd.

a)Show that the length of the of z is the product of the lengths of Z_{1}and Z_{2}.

b)Show that the angle between z and the x-axis is the sum of of the angles made by Z_{1}and Z_{2}

2. Relevant equations

tan(θ_{1})=b/a

tan(θ_{2})=d/c

|Z_{1}|=Z_{1}

|Z_{2}|=Z_{2}

3. The attempt at a solution

So the first part I didn't have any trouble with, it was fairly straight forward showing that the length of Z_{1}*Z_{2}was equal to the length of Z. But when I moved to part B I ran into a problem. Here's what I tried.

I Knew θ_{1}=b/a and θ_{2}=d/c by a first order taylor expansion of the tangents of these angles and since I am told the angle of Z, θ_{Z}is the sum of these two. I must prove:

θ_{Z}=(cb+da)/ca

However when I write out the form of Z=Z_{1}Z_{2}I get:

Z=ac-bd +j(ad+bc). Now since I know the tan(θ_{Z})=imaginary/real

I get tan(θ_{Z})=(ad+bc)/(ac-bd).

I'm not quite sure what I'm doing wrong, but I'd really appreciate a hand! Thanks!

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# Homework Help: Complex Vector Question

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