1. The problem statement, all variables and given/known data
Salve! In "The Mechanical Universe" introductory physics course, the ninth lecture called "Moving in Circles" shows the following vector equation, 2. Relevant equations

where i and j are vector indicators (with the little hat over them lol i=cartesian x, j=y)

3. The attempt at a solution
My question is that they display a circle and the r from the centre (radius) is spinning counter-clockwise and as it does so something strange happens. The show displays the above equation beside the circle and as the radius spins round the

or the

begin to spin around too, in what looks to me like a pattern trigonometry with an explanation. My suspicion is that there is a reason for this lol and i assume it has to do with the following pattern i noticed;

When the r points down south and proceeds spinning right, until it reaches up top (north) "cos w t" is readable and "sin w t" is upside down, then when it goes from north to south in the counter clockwise direction "cos w t" is upside down and "sin w t" is right side up. Now I wonder if this is to do with using "- sin w t" or is there an explanation somebody could give me, in laymans AND technical terms, just to quantify my fritterish thoughts. It'd really be a big help if someone would be able to help.

if my description is too ridiculous to follow you can see what i'm talking about at this link;
http://video.google.com/videoplay?docid=268446297904859913 [Broken]
at the 12.00 minute mark.

amabo te :) 1. The problem statement, all variables and given/known data

Don't read too deep into it, heh. You'll notice that the cosine and sine notations turn into flat lines, when their respective values are 0, and are completely spread out when their respective values are 1.

Really though, don't stress it. As long as you understand the axial projections of circular motion are rcos(wt) and rsin(wt), you're good to go. Have fun with SHM. :)

Well, the words "sin wt" and "cos wt" turning upside down (I wondered about that until is looked at the clip) is just cute graphics. It does have some meaning though- each is upside down when its value is negative. If you were to draw graphs of y= sin(wx) and y= cos(wx) you would see the same shape with one "trailing behind"- when cos(wx) is 1, sin(wx) is 0, as cos(x) drops to 0, sin(x) rises to one, as cos(x) drops to -1, sin(x) drops to 0, etc. That is because [math]sin^2(x)+ cos^2(x)= 1[/math] for all x. And, of course, that is why sine and cosin can be used as x and y components of a vector equations for a circle. If x= cos(wt) and y= sin(wt) then [itex]cos^2(wt)+ sin^2(wt)= x^2+ y^2= 1[/math], the equation of a circle with center at (0,0) and radius 1.