Winners and Answer to the Win SBD in this Math Contest #1, Simplify

The results are in for the Win SBD in this Math Contest #1

I hope you all had some fun with this.

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The simplify problem:
On the (x,y) plane, simplify:     u_xi   =   ?

u_x  =   unit vector in the x direction
i²   =   -1

One way to do a little research on this is to watch the very first post I made on steemit about 8 months ago, a 10 minute video called 'Imaginary Numbers Breakthrough to Reality'. (I suggested that in the Hint I posted a few days ago.) https://steemit.com/math/@point/update-and-hint-on-win-sbd-in-this-math-contest-1-simplify

If you watched the video you would see why:

u_xu_y   =   i

replace the i in the simplify problem and you have:

u_xi   =   u_xu_xu_y   =   (1)u_y   =   u_y

You can see, using simple algebra, why i rotates a vector by 90^o

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This algebra was discovered in the 1800's by Hermann Grassmann, who developed Exterior Algebra and further by William Clifford who developed Geometric Algebra. This has not been widely used or taught because of the popularity of Gibb's Vector Algebra. Since the 1960's Geometric Algebra has been promoted and further extended by David Hestenes of Arizona State University. But Geometric Algebra, although more and more researchers are using it, it is still not widely used by most researchers.

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The difference between a positive rotation and a minus rotation is just a matter of definition, just like deciding that electrons will be called negative and protons positive. Now through wide agreement it has been established for many years that a positive rotation is in the counter-clockwise direction, from the x axis up to the y axis. There is nothing inherent in the math that requires this choice, but if people don't agree there will be much confusion.

So a positive i rotates in the positive direction (counter-clockwise) and a negative i in the negative direction (clockwise).

Knowing that +/- i is the product of any two perpendicular unit vectors can really help to understand complex numbers:   Z   =   a + i b

Look at the amazing thing that happens when you multiply through by the unit vector u_x

u_xZ   =   u_x(a + i b)   =   u_xa + u_xi b   =   u_xa + u_xu_xu_yb   =   u_xa + u_yb

The "imaginary" number i is gone and we now have a very simple real vector equation that is easy to graph in the real (x,y) plane.

The result is a real vector u_xZ with a length of (a² + b²)^0.5     (Pythagoras)

As you can see Z operated on vector u_x through rotating and changing it's length to give us a new real vector u_xZ in the new direction of arctan(b/a)

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The winners are:

First to post the correct answer is @math-trail
First to prove their answer is @rycharde
First to post the algebra that proves his correct answer is @happychau123
Every participant recieves a part of a SBD.

So the winners are:

2 SBD to @math-trail
3 SBD to @rycharde
3 SBD to @happychau123
0.33 SBD to @kenchung
0.33 SBD to @anyway
0.33 SBD to @punjammies

Congratulations to all the winners.
I will send these winnings as soon as I get this posted.

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Some of you might enjoy watching the Heart Equation in Motion here in this 5 minute video:

https://steemit.com/math/@point/motion-math-art-watch-the-heart-equation-become-almost-alive-with-some-awesome-motion

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I hope you all had fun playing with this contest. And hopefully some of you might even have learned a "real" way to play with the "imaginary" number i.

Thank you to all those who commented and upvoted.