Write z_1 = -sqrt2i, z_2 = -3-3sqrt3i in polar form. Find product, quotients.

[CJwfjbq]

Write z₁ and z₂in polar form. (Express θ in radians. Let 0 ≤ θ < 2π.)
z1 = −sqrt2i
z 2 = −3−3sqrt3i

Z1=?
Z2=?

Find the product z₁z₂ and the quotients z1/z2 and 1/z1. (Express your answers in polar form with θ expressed in radians.)

Z1Z2=?
Z1/Z2=?
1/Z1=?

 

[stapel]

a. Write\(\displaystyle \, z_1\, =\, -\sqrt{\strut 2\,}i\, \)and \(\displaystyle \, z_2\, =\, -3\, -\, 3\sqrt{\strut 3\,}i\, \) in polar form. (Express θ in radians. Let 0 ≤ θ < 2π.)

b. Find the product z₁z₂ and

c. the quotient \(\displaystyle \, \dfrac{z_1}{z_2}\, \) and

d. the quotient \(\displaystyle \, \dfrac{1}{z_1}.\)

What formula(s) did they give you for doing this conversion? How far have you gotten in applying that information?

Please be complete. Thank you!

 

[pka]

Write z₁ and z₂in polar form. (Express θ in radians. Let 0 ≤ θ < 2π.)
z1 = −sqrt2i
z 2 = −3−3sqrt3i ?

You clearly did not read and agree to the rules before posting. Where is your work?

Here is a start. If \(\displaystyle z=x+y\bf{i} \) then first find its argument, \(\displaystyle \theta\)
So if \(\displaystyle x\cdot y =0 \) then \(\displaystyle \theta=~0,~\frac{\pi}{2},~\pi,\text{ or }\frac{3\pi}{2} \)
You must learn to recognize each of those above,

Suppose that then now that \(\displaystyle x\cdot y\ne 0 \) then let \(\displaystyle \rho=\arctan\left(\left|\dfrac{y}{x}\right| \right)\).
\(\displaystyle \arg(x + y\bf{i})=\theta = \left\{ {\begin{array}{{rl}} {\rho,}&{x > 0\;\& \;y > 0} \\ {\pi-\rho ,}&{x < 0\;\& \;y > 0} \\ { \pi+\rho ,}&{x < 0\;\&\;y < 0}\\ { 2\pi-\rho ,}&{x > 0\;\&\;y < 0} \end{array}} \right. \)

 

[Prove It]

Write z₁ and z₂in polar form. (Express θ in radians. Let 0 ≤ θ < 2π.)
z1 = −sqrt2i
z 2 = −3−3sqrt3i

Z1=?
Z2=?

Find the product z₁z₂ and the quotients z1/z2 and 1/z1. (Express your answers in polar form with θ expressed in radians.)

Z1Z2=?
Z1/Z2=?
1/Z1=?

Is the first one \(\displaystyle \displaystyle \begin{align*} -\sqrt{2}\,\mathbf{i} \end{align*}\) or \(\displaystyle \displaystyle \begin{align*} -\sqrt{2\,\mathbf{i}} \end{align*}\)?