180k views
22 votes
Consider the complex numbers and in the complex plane below:

Find the rectangular forms of v,w,z
and enter them in that order.

Consider the complex numbers and in the complex plane below: Find the rectangular-example-1
User ZAT
by
4.9k points

1 Answer

5 votes

Answer:

  • v = √2 +√2i
  • w = -2 -2√3i . . . ("i" is not under either radical)
  • z = 0 -3i

Explanation:

A suitable calculator can perform polar to rectangular conversion for you. If you like to do it "by hand", the conversion is ...

(r; θ) ⇔ r·cos(θ)+ r·sin(θ)i

Your numbers translate to ...

  • v = √2 +√2i
  • w = -2 -2√3i
  • z = 0 -3i

_____

Additional comment

We aren't often asked to read polar graphs. The concentric rings here are taken to be 1 unit apart. The radial lines are 15° apart. Then the various points are ...

v = 2∠45°

w = 4∠-120°

z = 3∠-90°

As you can see, the TI-84 work-alike in the attachment is able to write and understand numbers in this form. Other calculators treat complex numbers as vectors [magnitude, angle], or as coordinates with a special separator (semicolon) to indicate polar coordinates vs. complex plane coordinates: (r; θ) vs. x +iy

In any event, you need to be careful to understand the units the calculator uses for angles. In the attached, we have set the angle units to degrees. Most spreadsheets default to radians.

Consider the complex numbers and in the complex plane below: Find the rectangular-example-1
User Muhammad Noman
by
4.9k points