Angle-Vector Problem: Solving for the Angle Between Two Vectors z1 and z2

[JJHK]

Homework Statement

Consider a vector z defined by the equation z = z₁z₂, where z₁ = a + jb, z₂ = c + jd

Show that the angle between z and the x-axis is the sum of the angles made by z₁ and z₂ separately.

(EDIT): PLEASE LOOK AT MY BELOW POST!

 

[tiny-tim]

Hi JJHK!

show us what you've tried, and where you're stuck, and then we'll know how to help!

 

[JJHK]

Okay, well here's actually the first part of the problem. I also forgot to define what j is. It is an instruction to perform a counterclockwise rotation of 90°.

Now here's the first part:

Consider a vector z defined by the equation z = z1z2, where z1 = a + jb, z2 = c + jd

(a) Show that the length of z is the product of the lengths of z1 and z2

I first found the length of z1 and z2, L(z1) and L(z2):

L(z1) = √(a²+b²)

L(z2) = √(c²+d²)

Now I'm going to find the length of z, L(z):

L(z) = √((ac-bd)²+(ad+bc)²)
= √(a²c²+a²d²+b²c²+b²d²)

I'm going to show that the above solution is equal to L(z1)L(z2)

L(z1)L(z2) = √(a²+b²)√(c²+d²)
= √(a²c²+a²d²+b²c²+b²d²)

Therefore, L(z) = L(z1)L(z2)

Now part 2:

(b)Show that the angle between z and the x-axis is the sum of the angles made by z1 and z2 separately.

I first found the angles of z1 and z2, θ(z1) and θ(z2):

θ(z1) = arctan(b/a) and θ(z2) = arctan(d/c)

and the angle for z is:

θ(z) = arctan((ad+bc)/(ac-bd))

Now I'm stuck, how do I equate θ(z) = θ(z1) + θ(z2) ??

THanks for the help!

 

[tiny-tim]

so far so good!

hint: if tan^-1A = tan^-1B + tan^-1C,

then A = tan(tan^-1B + tan^-1C) …

and what's the formula for tan of a sum? ​