Multipying/Dividing Vectors what do you do with their Angles?

For the record, I'll answer my own question:

let: ε = the base of natural logarithyms.

Let: j = squareroot[-1]

Euler's formula [see: http://en.wikipedia.org/wiki/Euler's_formula ] says:

ε^(jx) = cosx + jsinx

Therefore two vectors:

A(cosφ + jsinφ) = A*ε^(jφ)

B(cosψ + jsinψ) = B*ε^(jψ)

so for multiplication:

[A(cosφ + jsinφ)]*[B(cosψ + jsinψ)]

= [A*ε^(jφ)]*[B*ε^(jψ)] = (A*B)*ε^(jφ+jψ)

= (A*B)*ε^j(φ+ψ)

by Euler's rule you can expand that to

(A*B)*[cos(φ+ψ) + jsin(φ+ψ)]

and for division:

[A(cosφ + jsinφ)]/[B(cosψ + jsinψ)]

= [A*ε^(jφ)]/[B*ε^(jψ)] = (A/B)*ε^(jφ-jψ)

= (A/B)*ε^j(φ-ψ)

by Euler's rule you can expand that to

(A/B)*[cos(φ-ψ) + jsin(φ-ψ)]

And that's the basis for what I was taught (and you may have learned) by rote:

When multiplying vectors you add their angles.

When dividing vectors subtact the angle of the denominator from the angle of the numerator.

It's really as simple as Advanced Algebra. :)