Show the length of vector z is the product of z1 and z2.

[bfusco]

Homework Statement

consider a vector z defined by the equation z=z1z2, where z1=a+jb, and z2=c+jd (j=complex: same as 'i').
(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.

P.S. this question is part of a personal study, so feel free to bring all information applicable without skipping steps, as to really solidify my understanding because i have to admit I've gotten to these 'upper level' math and physics classes and i feel as though i know nothing. lol

Thank you in advance!

The Attempt at a Solution

first off considering that this is 2 vectors being multiplied to get another vector I am assuming that this is the cross product. if that is the case than i would start by simply taking the determinant resulting in: ad-cb, but at i wouldn't know what to do past this point.

 

[Mark44]

Homework Statement

consider a vector z defined by the equation z=z1z2, where z1=a+jb, and z2=c+jd (j=complex: same as 'i').
(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.

P.S. this question is part of a personal study, so feel free to bring all information applicable without skipping steps, as to really solidify my understanding because i have to admit I've gotten to these 'upper level' math and physics classes and i feel as though i know nothing. lol

Thank you in advance!

The Attempt at a Solution

first off considering that this is 2 vectors being multiplied to get another vector I am assuming that this is the cross product. if that is the case than i would start by simply taking the determinant resulting in: ad-cb, but at i wouldn't know what to do past this point.

I don't believe that the cross product is called for here, and I also believe that the description of z as a vector is somewhat misleading. You are given that z₁ and z₂ are complex numbers, for which ordinary multiplication is defined.

 

[Dick]

It's really pretty easy if you express the complex numbers in polar form, if you gotten that far and want a shortcut.