Section+2.4+9-23

Section 2.4

Last Friday we compared the usefulness of Cartesian (rectangular) vs Polar coordinates.

We came up with the following list


 * Polar coordinates- rcis(****θ)**
 * -Easier to Multiply**
 * -Easier to find the Argument**
 * -Easier to find the Magnitude**


 * Cartesian Coordinates- (X+Yi)**
 * -Easier to Add**
 * -Easier to graph (due to familiarity)**

We then proceeded to use various examples to prove one thing or the other. To prove that it is easier to find the Argument and Magnitude with Polar Coordinates we used Problem 8 on the homework. Its really only necessary to show 8a and 8b to get the point across. find the magnitude of the and the argument of this equation.
 * 8a**

{ something is not right here... the argument is not in the right quadrant and somehow the sqrt(3) jumped into the numerator of the fraction} find the magnitude and the argument of this equation
 * 8b**

(not even gonna need code cogs for this)

Magnitude=10 Argument=11 π/6

(don't believe me? check page 94 chapter 2)

So we know that its easier to find the magnitude and Argument using Polar coordinates but how about why its easier to multiply. Now we have to be hardcore and use graphs to prove this. The graph above is from Geogebra. It demonstrates that you can multiply a complex number z by k by stretching the vector by a factor of k.

The graph above is also From Geogebra. It demonstrates that you can add complex numbers by completing a parallelogram with the vectors. The graph above is from (you guessed it good job :D) Geogebra (if you didn't guess that than you need more sleep ʘ‿ʘ ) It demonstrates that multiplying a complex number z by i will rotate it by π/2 counter clockwise. The graph above is from Wolfram Alpha (just trolling its from Geogebra). The equation for the graph is below. in particular look at the right side of the equation. View 4+i as W and follow the first three graphs on how to graph this. You end up with the graph above. However if you simplify this before you graph it you end up with. The graph of which below does not tell you anything very interesting. By carefully observing graph number 4 (the rectangular one) you can obtain a rather useful theorem
 * THEOREM @>@ (2.2)**
 * Given complex numbers z=acis****α** **and w =bcis**β

If you want proof look at page 100-101 in your text book.

Your fellow student Daniel Lian (PS this is not a photo of me i look more like this [])