How Do Rotational Transformations Affect Coordinates?

[Fusilli_Jerry89]

Just a quick question:

In my prof's lecture he drew two rotated rulers of the same length and showed that length is invariant under rotations. Obviously x^2 + y^2 = x'^2 + y'^2 = length of ruler, but how do we go from this to:

x' = xcos(theta) + ysin(theta)
y' = ycos(theta) - xsin(theta)

That's what my prof wrote down..

 

[tiny-tim]

In my prof's lecture he drew two rotated rulers of the same length and showed that length is invariant under rotations. Obviously x² + y² = x'² + y'² = length of ruler, but how do we go from this to:

x' = xcosθ + ysinθ
y' = ycosθ - xsinθ

Hi Fusilli_Jerry89!

(have a theta: θ and a squared: ² )

Just draw a circle with two rulers in it, one along the x-axis, and the other at an angle θ.

Then use trig.

 

[Fusilli_Jerry89]

lol i sorry I know this is easy but for some reason I can't figure out where we get the y from. Is that a component? And is it the hypotenuse of something?

 

[tiny-tim]

lol i sorry I know this is easy but for some reason I can't figure out where we get the y from. Is that a component? And is it the hypotenuse of something?

No, y is the other short side of the triangle.

Yes, y is a component, just like x.

The ruler in the circle at an angle θ has its endpoint at x' = 1, y' = 0,

and simple trig shows that that is the same as x = cosθ, y = sinθ.