# A simple angle-vector problem!

1. Feb 28, 2012

### JJHK

1. The problem statement, all variables and given/known data

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!!

Last edited: Feb 28, 2012
2. Feb 28, 2012

### tiny-tim

Hi JJHK!

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

3. Feb 28, 2012

### 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):

= √(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:

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

THanks for the help!

4. Feb 28, 2012

### 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?